The Greatest
Mathematicians of the Pastranked in approximate order of "greatness."
To qualify, the mathematician
must be born before 1930 and his work must have
breadth,
depth, and
historical importance.
(Fifty-five additional names to change this List from 'Greatest of the Past'
to 'Greatest of All Time appear after the first 145 names.)

In compiling this list, I've considered contributions
outside mathematics.
I already give lower weight to breadth and influence in
mathematical physics, but if I reduced the weight to zero,
the List would be much different.
Newton contributed little to number theory,
but is considered to have breadth because of his physics,
which is also his main influence. If only breadth and influence
in pure mathematics
are considered, Newton wouldn't be #1 (though still in the Top Ten).

Einstein, Galileo, Maxwell, Aristotle and Cardano are among the greatest
applied mathematicians in history, but lack the importance as
pure mathematicians to qualify for The Top 70.
Nevertheless I'd want to include them in any longer list, so
I've tucked these ambiguous cases into the #71-#75 slots.

The Top 100 represent a list of
Greatest Mathematicians of the Past,
with 1930 birth as an arbitrary cutoff, but there are at least
five mathematicians born after 1930 who would surely belong on the
Top 100 list were this date restriction lifted.
I've shown them in the #101-#105 slots.
Another five mathematicians born after 1930 probably belong
at least in the Top 150 and I show them in the #146-#150 slots.
However, to keep the focus on Mathematicians of the Past,
these are the only ten mathematicians
born post-1930 who are shown in the Top 150.

As mentioned above, ranks #146-150 are reserved,
along with #101-105, for
the ten greatest mathematicians born after 1930.

... And I still couldn't stop! :-)
I've expanded the List to 200 names, mostly with 20th-century
mathematicians (including 17 more born 1930 or later)..
I don't try to "rank" these final fifty
(though having two tiers is convenient).
I've not written mini-bios for most of these, but have linked
(with a slightly different color of link) to the excellent
biographies at Wikipedia or the MacTutor site.
Please note that, although I've placed them here
after the "Top 150" some of these recent mathematicians
may belong among the Top 100 or the Top 150.

Since I am especially unqualified to rank modern mathematicians,
here are 15 more born in the 20th century whom I didn't squeeze
onto the List of 200, but who might belong there anyway:
Bombieri
Bourgain
Fefferman
Hirzebruch
Kontsevich
Manin
Margulis
Mazur
Nirenberg
Novikov
Quillen
J.Robinson
W.Werner
Wiles
Yau

Click for a
discussion of certain omissions.
Please send me e-mail if
you believe there's a major flaw in
my rankings (or an error in any of the biographies).
Obviously the relative ranks of, say Fibonacci and
Ramanujan, will never satisfy everyone since the reasons
for their "greatness" are different.
I'm sure I've overlooked great mathematicians
who obviously belong on this list.
Please e-mail and tell me!

Following are the top mathematicians in
chronological (birth-year) order.
(By the way, the ranking assigned to a mathematician will appear if you
place the cursor atop the name at the top of his mini-bio.):

Earliest mathematicians

Little is known of the earliest mathematics, but the famous
Ishango Bone from Early Stone-Age Africa has tally marks suggesting
arithmetic. The markings include six prime numbers (5, 7, 11, 13, 17, 19)
in order, though this is probably coincidence.

The advanced artifacts of Egypt's Old Kingdom
and the Indus-Harrapa civilization
imply strong mathematical skill, but the first
written evidence of advanced arithmetic dates from Sumeria,
where 4500-year old clay tablets show multiplication and
division problems; the first abacus may be about this old.
By 3600 years ago, Mesopotamian tablets show
tables of squares, cubes, reciprocals, and even logarithms
and trig functions,
using a primitive place-value system (in base 60, not 10).
Babylonians were familiar with the Pythagorean Theorem,
solutions to quadratic equations,
even cubic equations (though they didn't have a general solution for these),
and eventually even developed methods to estimate
terms for compound interest.
The Greeks borrowed from Babylonian mathematics, which was the most
advanced of any before the Greeks; but there is no
ancient Babylonian mathematician whose name is known.

Also at least 3600 years ago, the Egyptian scribe Ahmes
produced a famous manuscript
(now called the Rhind Papyrus),
itself a copy of a late Middle Kingdom text.
It showed simple algebra methods and
included a table giving optimal
expressions using Egyptian fractions.
(Today, Egyptian fractions lead to challenging number theory
problems with no practical applications, but they may have
had practical value for the Egyptians.
To divide 17 grain bushels among 21 workers, the
equation 17/21 = 1/2 + 1/6 + 1/7 has practical value,
especially when compared with
the "greedy" decomposition
17/21 = 1/2 + 1/4 + 1/17 + 1/1428.)

The Pyramids demonstrate that Egyptians
were adept at geometry, though little written evidence survives.
Babylon was much more advanced than Egypt at arithmetic and algebra;
this was probably due, at least in part, to their place-value system.
But although their base-60 system survives (e.g. in the division
of hours and degrees into minutes and seconds) the Babylonian
notation, which used the equivalent of IIIIII XXXXXIIIIIII XXXXIII
to denote 417+43/60, was unwieldy compared to the
"ten digits of the Hindus."
(In 2016 historians were surprised to decode ancient Babylonian
texts and find very sophisticated astronomical calculations
of Jupiter's orbit.)

The Egyptians used the approximation
π ≈ (4/3)4
(derived from the idea that a
circle of diameter 9 has about the same area as a square of side 8).
Although the ancient Hindu mathematician Apastambha had achieved
a good approximation for √2, and the ancient Babylonians
an ever better √2, neither of these ancient
cultures achieved a π approximation as good as Egypt's,
or better than π ≈ 25/8, until the Alexandrian era.

The greatest mathematics before the
Golden Age of Greece was in India's
early Vedic (Hindu) civilization.
The Vedics understood relationships between geometry
and arithmetic, developed astronomy, astrology, calendars,
and used mathematical forms in some religious rituals.

The earliest mathematician to whom definite teachings can be ascribed
was Lagadha, who apparently lived about 1300 BC and
used geometry and elementary trigonometry for his astronomy.
Baudhayana lived about 800 BC and also wrote on algebra and geometry;
Yajnavalkya lived about the same time and is credited with the then-best
approximation to π.
Apastambha did work summarized below;
other early Vedic mathematicians solved quadratic and simultaneous equations.

Other early cultures also developed some mathematics.
The ancient Mayans apparently had a place-value system
with zero before the Hindus did;
Aztec architecture implies practical geometry skills.
Ancient China certainly developed mathematics,
in fact the first known proof of the Pythagorean Theorem
is found in a Chinese book (Zhoubi Suanjing)
which might have been written about 1000 BC.

Thales was the Chief of the "Seven Sages" of ancient Greece,
and has been called the "Father of Science,"
the "Founder of Abstract Geometry,"
and the "First Philosopher."
Thales is believed to have studied mathematics
under Egyptians, who in turn were aware of much older
mathematics from Mesopotamia.
Thales may have invented the notion of compass-and-straightedge
construction.
Several fundamental theorems about triangles
are attributed to Thales, including the law of similar triangles
(which Thales used famously to calculate the height of the Great Pyramid)
and "Thales' Theorem" itself: the fact that any angle inscribed in
a semicircle is a right angle.
(The other "theorems" were probably more like well-known axioms,
but Thales proved Thales' Theorem using two of
his other theorems; it is said that Thales then sacrificed
an ox to celebrate what might have been the first
mathematical proof in Greece.)
Thales noted that, given a line segment
of length x, a segment of length x/k can be constructed
by first constructing a segment of length kx.

Thales was also an astronomer; he invented the 365-day calendar,
introduced the use of Ursa Minor for finding North,
invented the gnomonic map projection (the first of many
methods known today to map (part of) the surface of a sphere
to a plane,
and is the first person believed to have correctly predicted a solar eclipse.
His theories of physics would seem quaint today, but he
seems to have been the first to describe magnetism and static electricity.
Aristotle said, "To Thales the primary question was
not what do we know, but how do we know it."
Thales was also a politician, ethicist, and military strategist.
It is said he once leased all available olive presses
after predicting a good olive season; he did this not
for the wealth itself, but as a demonstration of the use
of intelligence in business.
Thales' writings have not survived and are known only second-hand.
Since his famous theorems of geometry were probably already known
in ancient Babylon, his importance derives from imparting the
notions of mathematical proof and the scientific method
to ancient Greeks.

Thales' student and successor was Anaximander, who is often called
the "First Scientist" instead of Thales: his theories were more
firmly based on experimentation and logic, while Thales still
relied on some animistic interpretations.
Anaximander is famous for astronomy, cartography and sundials,
and also enunciated a theory of evolution, that land species
somehow developed from primordial fish!
Anaximander's most famous student, in turn, was Pythagoras.
(The methods of Thales and Pythagoras led to the schools of
Plato and Euclid, an intellectual blossoming unequaled until
Europe's Renaissance.
For this reason Thales may belong on this list for his
historical importance despite his relative lack of
mathematical achievements.)

The Dharmasutra composed by Apastambha contains mensuration
techniques, novel geometric construction techniques, a method of elementary
algebra, and what may be an early proof of the Pythagorean Theorem.
Apastambha's work uses the excellent (continued fraction)
approximation √2
≈ 577/408, a result probably derived with
a geometric argument.

Apastambha built on the work of earlier Vedic scholars,
especially Baudhayana, as well
as Harappan and (probably) Mesopotamian mathematicians.
His notation and proofs were primitive, and there is little
certainty about his life.
However similar comments apply to Thales of Miletus, so it seems
fair to mention Apastambha (who was perhaps the
most creative Vedic mathematician before Panini)
along with Thales as one of the
earliest mathematicians whose name is known.

Pythagoras, who is sometimes called the "First Philosopher,"
studied under Anaximander, Egyptians, Babylonians,
and the mystic Pherekydes (from whom
Pythagoras acquired a belief in reincarnation); he became
the most influential of early Greek mathematicians.
He is credited with being first to use axioms
and deductive proofs, so his influence on Plato and Euclid may be enormous;
he is generally credited with much of
Books I and II of Euclid's Elements.
He and his students (the "Pythagoreans") were ascetic mystics
for whom mathematics was partly a spiritual tool.
(Some occultists treat Pythagoras as a wizard and founding mystic
philosopher.)
Pythagoras was very interested in astronomy and seems to have been the
first man to realize that the Earth was a globe similar to the other planets.
He and his followers began to study the question of planetary motions,
which would not be resolved for more than two millennia.
He believed thinking was located in the brain rather than heart.
The words philosophy and mathematics are said to have
been coined by Pythagoras.
He is supposed to have invented the Pythagorean Cup, a clever
wine goblet which punishes a drinker who greedily fills his cup to the top
by then using siphon pressure to drain the cup.

Despite Pythagoras' historical importance I may have ranked him too high:
many results of the Pythagoreans were
due to his students; none of their writings survive; and what
is known is reported second-hand, and possibly
exaggerated, by Plato and others.
Some ideas attributed to him were probably first enunciated by successors
like Parmenides of Elea (ca 515-440 BC).
Archaeologists now believe that he was not first to invent the diatonic
scale:
Here is a diatonic-scale song from Ugarit
which predates Pythagoras by eight centuries.

Pythagoras' students included Hippasus of Metapontum,
the famous anatomist and physician Alcmaeon, Milo of Croton,
and Croton's daughter Theano (who may have been Pythagoras's wife).
The term Pythagorean was also adopted by many disciples who lived later;
these disciples include Philolaus of Croton,
the natural philosopher Empedocles, and several other famous Greeks.
Pythagoras' successor was apparently Theano herself:
the Pythagoreans were one of the few ancient schools
to practice gender equality.

Pythagoras discovered that harmonious intervals in music are based on
simple rational numbers.
This led to a fascination with integers
and mystic numerology;
he is sometimes called the "Father of Numbers"
and once said "Number rules the universe."
(About the mathematical basis of music, Leibniz later wrote,
"Music is the pleasure the human soul experiences from counting
without being aware that it is counting."
Other mathematicians who investigated the arithmetic
of music included Huygens, Euler and Simon Stevin.)
Given any numbers a and b the Pythagoreans were aware
of the three distinct means:
(a+b)/2 (arithmetic mean),
√(ab) (geometric mean), and
2ab/(a+b) (harmonic mean).

The Pythagorean Theorem was known long before Pythagoras,
but he was often credited (before discovery of an ancient Chinese text)
with the first proof.
He may have discovered the simple parametric form
of primitive Pythagorean triplets (xx-yy, 2xy, xx+yy),
although the first explicit mention of this may be in Euclid's Elements.
Other discoveries of the Pythagorean school include the
construction of the regular pentagon,
concepts of perfect and amicable numbers,
polygonal numbers,
golden ratio (attributed to Theano),
three of the five regular solids (attributed to Pythagoras himself),
and irrational numbers (attributed to Hippasus).
It is said that the discovery
of irrational numbers upset the Pythagoreans so much
they tossed Hippasus into the ocean!
(Another version has Hippasus banished for revealing
the secret for constructing the sphere which circumscribes
a dodecahedron.)

In addition to Parmenides, the famous successors of Thales and Pythagoras
include Zeno of Elea (see below),
Hippocrates of Chios (see below),
Plato of Athens (ca 428-348 BC),
Theaetetus (see below), and Archytas (see below).
These early Greeks ushered in a Golden Age of Mathematics
and Philosophy
unequaled in Europe until the Renaissance.
The emphasis was on pure, rather than practical, mathematics.
Plato (who ranks #40 on Michael Hart's famous list of
the Most Influential Persons in History)
decreed that his scholars should do
geometric construction solely with compass and straight-edge
rather than with "carpenter's tools" like
rulers and protractors.

Panini's great accomplishment was his study
of the Sanskrit language, especially in his text Ashtadhyayi.
Although this work might be considered the very first study
of linguistics or grammar, it used a non-obvious elegance
that would not be equaled in the West until the 20th century.
Linguistics may seem an unlikely qualification for a "great mathematician,"
but language theory is a field of mathematics.
The works of eminent 20th-century linguists and computer scientists
like Chomsky, Backus, Post and Church
are seen to resemble Panini's work 25 centuries earlier.
Panini's systematic study of Sanskrit may have inspired
the development of Indian science and algebra.
Panini has been called "the Indian Euclid" since the rigor
of his grammar is comparable to Euclid's geometry.

Although his great texts have been preserved, little else
is known about Panini. Some scholars would place his dates a century
later than shown here; he may or may not have been the same person
as the famous poet Panini.
In any case, he was the very last Vedic Sanskrit scholar by
definition: his text formed the transition to the
Classic Sanskrit period.
Panini has been called "one of the most innovative
people in the whole development of knowledge;"
his grammar "one of the greatest monuments of human intelligence."

Zeno, a student of Parmenides, had great fame in
ancient Greece.
This fame, which continues to the present-day, is largely due to
his paradoxes of infinitesimals, e.g. his argument
that Achilles can never catch the tortoise (whenever Achilles arrives
at the tortoise's last position, the tortoise has moved on).
Although some regard these paradoxes as simple fallacies,
they have been contemplated for many centuries.
It is due to these paradoxes that
the use of infinitesimals, which provides the basis for mathematical analysis,
has been regarded as a non-rigorous heuristic and
is finally viewed as sound only after the work of the great
19th-century rigorists, Dedekind and Weierstrass.
Zeno's Arrow Paradox (at any single instant an arrow is at a fixed
position, so where does its motion come from?) has lent its name to
the Quantum Zeno Effect, a paradox of quantum physics.

Eubulides of Miletus
was another ancient Greek famous for paradoxes,
e.g. "This statement is a lie" -- the sort of inconsistency
later used in proofs by Gödel and Turing.

Hippocrates (no known relation to Hippocrates of Cos,
the famous physician)
wrote his own Elements more than
a century before Euclid. Only fragments survive but it
apparently used axiomatic-based proofs similar to Euclid's
and contains many of the same theorems. Hippocrates is said to
have invented the reductio ad absurdem proof method.
Hippocrates is most famous for his work on the three ancient
geometric quandaries: his work on cube-doubling
(the Delian Problem) laid the groundwork
for successful efforts by Archytas and others;
and some claim Hippocrates was first to trisect the general angle.
His circle quadrature was of course ultimately unsuccessful but
he did prove ingenious theorems about "lunes" (crescent-shaped
circle fragments).
For example, the area of any right triangle is equal to the
sum of the areas of the two lunes formed when semi-circles
are drawn on each of the three edges of the triangle.
Hippocrates also did work in algebra and rudimentary analysis.

(Doubling the cube and angle trisection are often called
"impossible," but they are impossible only when restricted to
collapsing compass and unmarkable straightedge.
There are ingenious solutions available with other tools.
Construction of the regular heptagon is another such task,
with solutions published by four of the men on this List:
Thabit, Alhazen, Vieta, Conway.)

Archytas was an important statesman as well
as philosopher. He studied under Philolaus of Croton,
was a friend of Plato, and tutored Eudoxus.
In addition to discoveries always attributed to him,
he may be the source of several of Euclid's theorems,
and some works attributed to Eudoxus and perhaps Pythagoras.
Recently it has been shown that the magnificent Mechanical Problems
attributed to (pseudo-)Aristotle were probably actually written
by Archytas, making him one of the greatest mathematicians
of antiquity.

Archytas introduced "motion" to geometry, rotating curves to
produce solids.
If his writings had survived he'd surely be considered one
of the most brilliant and innovative geometers of antiquity.
He appears on Cardano's List of 12 Greatest Geniuses.
(Euclid, Aristotle, Archimedes, Apollonius, Ptolemy,
and the physician Galen of Pergamum
are the other Greeks on that List.)
Archytas' most famous mathematical achievement was
"doubling the cube" (constructing a line segment larger than
another by the factor cube-root of two).
Although others solved the problem with
other techniques, Archytas' solution for cube doubling was astounding because
it wasn't achieved in the plane, but involved the intersection
of three-dimensional bodies.
This construction (which introduced the Archytas Curve)
has been called "a tour de force of the spatial imagination."
He invented the term harmonic mean and worked with geometric means
as well (proving that consecutive integers never have rational geometric mean).
He was a true polymath: he advanced the theory of music far beyond Pythagoras;
studied sound, optics and cosmology;
invented the pulley (and a rattle to occupy infants); wrote about
the lever; developed the curriculum called quadrivium;
is credited with inventing the screw;
and is supposed to have built a steam-powered wooden bird which
flew for 200 meters.
Archytas is sometimes called the "Father of Mathematical Mechanics."

Some scholars think Pythagoras and Thales are
partly mythical. If we take that view, Archytas (and Hippocrates)
should be promoted in this list.

Theaetetus is presumed to be the true author
of Books X and XIII of Euclid's Elements, as well as some
work attributed to Eudoxus. He was considered one of the brightest
of Greek mathematicians, and is the central character
in two of Plato's Dialogs.
It was Theaetetus who discovered the final two of the five "Platonic solids"
and proved that there were no more.
He may have been first to note that the square root of any integer,
if not itself an integer, must be irrational.
(The case √2 is attributed to a student of Pythagoras.)

Eudoxus journeyed widely for his education,
despite that he was not wealthy,
studying mathematics with Archytas in Tarentum,
medicine with Philiston in Sicily,
philosophy with Plato in Athens,
continuing his mathematics study in Egypt,
touring the Eastern Mediterranean with his own students
and finally returned to Cnidus where he established himself
as astronomer, physician, and ethicist.
What is known of him is second-hand, through the writings
of Euclid and others, but he was one
of the most creative mathematicians of the ancient world.

Many of the theorems
in Euclid's Elements were first proved by Eudoxus.
While Pythagoras had been horrified by the discovery of irrational
numbers, Eudoxus is famous for incorporating them into arithmetic.
He also developed the earliest techniques of the infinitesimal calculus;
Archimedes credits Eudoxus with inventing a principle eventually called
the Axiom of Archimedes:
it avoids Zeno's paradoxes by, in effect, forbidding
infinities and infinitesimals.
Eudoxus' work with irrational numbers, infinitesimals and limits
eventually inspired masters like Dedekind.
Eudoxus also introduced an Axiom of Continuity;
he was a pioneer in solid geometry;
and he developed his own solution to the Delian cube-doubling problem.
Eudoxus was the first great mathematical astronomer;
he developed the complicated ancient theory of planetary orbits;
and may have invented the astrolabe.
He may have invented the 365.25-day calendar based on leap
years, though it remained for Julius Caesar to popularize it.
(It is sometimes said that he knew that
the Earth rotates around the Sun, but that appears to be false;
it is instead Aristarchus of Samos, as cited by Archimedes, who
may be the first "heliocentrist.")
One of Eudoxus' students was Menaechmus, who was first to
describe the conic sections and used them to devise a non-Platonic
solution to the cube-doubling problem (and perhaps the
circle-squaring problem as well).

Four of Eudoxus' most famous discoveries were the volume
of a cone, extension of arithmetic to the irrationals, summing
formula for geometric series, and
viewing π as the limit of polygonal perimeters.
None of these seems difficult today, but it does seem remarkable
that they were all first achieved by the same man.
Eudoxus has been quoted as saying
"Willingly would I burn to death like Phaeton, were this
the price for reaching the sun and learning its shape,
its size and its substance."

Aristotle was the most prominent scientist of the
ancient world, and perhaps the most influential philosopher
and logician ever;
he ranks #13 on Michael Hart's list of the Most Influential Persons in History.
His science was a standard curriculum for almost 2000 years.
Although the physical sciences couldn't advance until the
discoveries by great men like Newton and Lavoisier,
Aristotle's work in the biological sciences
was superb, and served as paradigm until modern times.
Aristotle was personal tutor to the young Alexander the Great.

Although Aristotle was probably the greatest biologist of the ancient world,
his work in physics and mathematics may not seem enough
to qualify for this list.
But his teachings covered a very wide gamut and dominated
the development of ancient science.
His writings on definitions, axioms and proofs may have influenced Euclid;
and he was one of the first mathematicians to write on the subject of infinity.
His writings include geometric theorems, some with proofs
different from Euclid's or missing from Euclid altogether; one of these
(which is seen only in Aristotle's work prior to Apollonius)
is that a circle is the locus of points whose distances
from two given points are in constant ratio.

A charge sometimes made against Aristotle is that his wrong
ideas held back the development of science.
But this charge is unfair; Aristotle himself stressed the importance
of ovbservation and experimentation, and to be ready to reject
old hypotheses and prepare new ones.
And even if, as is widely agreed, Aristotle's geometric theorems
were not his own work, his status as the most influential
logician and philosopher in all of history
makes him a strong candidate for the List.

Euclid of Alexandria (not to be confused with Socrates' student,
Euclid of Megara, who lived a century earlier),
directed the school of mathematics at the great
university of Alexandria. Little else is known for certain about his life,
but several very important mathematical achievements are credited to him.
He was the first to prove that there are infinitely
many prime numbers; he produced an incomplete proof of
the Unique Factorization Theorem (Fundamental Theorem of Arithmetic);
and he devised Euclid's algorithm for computing gcd.
He introduced the Mersenne primes
and observed that (M2+M)/2
is always perfect (in the sense of Pythagoras) if M is Mersenne.
(The converse, that any even perfect number has such a corresponding
Mersenne prime, was tackled by Alhazen and proven by Euler.)
His books contain many famous theorems, though much of the Elements
was due to predecessors like Pythagoras (most of Books I and II),
Hippocrates (Book III), Theodorus, Eudoxus (Book V),
Archytas (perhaps Book VIII) and Theaetetus.
Book I starts with an elegant proof that rigid-compass constructions
can be implemented with a collapsing compass.
(Given A, B, C, find CF = AB by first
constructing equilateral triangle ACD;
then use the compass to find E on AD with AE = AB;
and finally find F on DC with DF = DE.)
Although notions of trigonometry were not in use, Euclid's theorems
include some closely related to the Laws of Sines and Cosines.
Among several books attributed to Euclid are
The Division of the
Scale (a mathematical discussion of music), The Optics,
The Cartoptrics (a treatise on the theory of mirrors),
a book on spherical geometry, a book on logic fallacies,
and his comprehensive math textbook The Elements.
Several of his masterpieces have been lost, including
works on conic sections and other advanced geometric topics.
Apparently Desargues' Homology Theorem (a pair of triangles
is coaxial if and only if it is copolar) was proved in one
of these lost works; this is the fundamental theorem
which initiated the study of projective geometry.
Euclid ranks #14 on Michael Hart's famous list of
the Most Influential Persons in History.
The Elements introduced the notions of axiom and theorem;
was used as a textbook for 2000 years; and in fact is still the basis
for high school geometry, making
Euclid the leading mathematics teacher of all time.
Some think his best inspiration was recognizing that the
Parallel Postulate must be an axiom rather than a theorem.

There are many famous quotations about Euclid and his books.
Abraham Lincoln abandoned his law studies when he didn't
know what "demonstrate" meant and "went home to my father's house
[to read Euclid], and stayed there till I could give any proposition
in the six books of Euclid at sight.
I then found out what demonstrate means, and went back to my law studies."

Archimedes is universally acknowledged to be
the greatest of ancient mathematicians.
He studied at Euclid's school (probably after
Euclid's death), but his work far surpassed, and even leapfrogged,
the works of Euclid.
(For example, some of Euclid's more difficult theorems are easy
analytic consequences of Archimedes' Lemma of Centroids.)
His achievements are particularly impressive given the
lack of good mathematical notation in his day.
His proofs are noted not only for brilliance but for
unequaled clarity, with a modern biographer (Heath) describing
Archimedes' treatises as "without exception monuments of mathematical
exposition ... so impressive in their perfection as to create a
feeling akin to awe in the mind of the reader."
Archimedes made advances in number theory, algebra, and analysis,
but is most renowned for his many theorems
of plane and solid geometry.
He was first to prove Heron's formula for the area of a triangle.
His excellent approximation to √3 indicates that
he'd partially anticipated the method of continued fractions.
He developed a recursive method of representing large
integers, and was first to note the law of exponents,
10a·10b = 10a+b.
He found a method to trisect an arbitrary angle (using
a markable straightedge — the construction is impossible
using strictly Platonic rules).
One of his most remarkable and famous geometric results
was determining the area of a parabolic section, for which
he offered two independent proofs, one
using his Principle of the Lever,
the other using a geometric series.
Some of Archimedes' work survives only because Thabit ibn Qurra
translated the otherwise-lost Book of Lemmas; it contains
the angle-trisection method and several ingenious theorems
about inscribed circles.
(Thabit shows how to construct a regular heptagon; it may not be
clear whether this came from Archimedes, or was fashioned by Thabit
by studying Archimedes' angle-trisection method.)
Other discoveries known only second-hand include
the Archimedean semiregular solids reported by Pappus,
and the Broken-Chord Theorem reported by Alberuni.

Archimedes and Newton might be the two best geometers ever, but
although each produced ingenious geometric proofs, often they used
non-rigorous calculus to discover results, and then devised
rigorous geometric proofs for publication.
He used integral calculus
to determine the centers of mass of hemisphere and cylindrical
wedge, and the volume of two cylinders' intersection.
He also worked with various spirals, paraboloids of revolution, etc.
Although Archimedes didn't develop differentiation (integration's inverse),
Michel Chasles credits him (along with Kepler, Cavalieri, and Fermat,
who all lived more than 18 centuries later)
as one of the four who developed calculus before Newton and Leibniz.
(Although familiar with the utility of infinitesimals, he
accepted the "Theorem of Eudoxus" which bans them to avoid
Zeno's paradoxes. Modern mathematicians refer to that "Theorem" as
the Axiom of Archimedes.)

Archimedes was an astronomer (details of his discoveries are lost,
but it is likely he knew the Earth rotated around the Sun).
He was one of the greatest mechanists ever, discovering
Archimedes' Principle of Hydrostatics (a body partially or completely
immersed in a fluid effectively loses weight equal to the
weight of the fluid it displaces).
He developed the mathematical foundations underlying the
advantage of basic machines: lever, screw and compound pulley.
Although the screw was perhaps invented by Archytas,
and Stone-Age man (and even other animals) used the lever,
it is said that the compound
pulley was invented by Archimedes himself.
For these achievements he is often ranked ahead of Maxwell
to be called one of the three greatest physicists ever.
Archimedes was a prolific inventor:
in addition to inventing the compound pulley, he
invented the hydraulic screw-pump (called Archimedes' screw);
a miniature planetarium; and several
war machines -- catapult, parabolic mirrors to burn enemy ships,
a steam cannon, and 'the Claw of Archimedes.'
Some scholars attribute the Antikythera mechanism
to Archimedes.

His books include
Floating Bodies,
Spirals,
The Sand Reckoner,
Measurement of the Circle,
Sphere and Cylinder,
Plane Equilibriums,
Conoids and Spheroids,
Quadrature of Parabola,
The Book of Lemmas (translated and attributed by Thabit ibn Qurra),
various now-lost works (on Mirrors, Balances and Levers, Semi-regular Polyhedra,
etc.) cited by Pappus or others,
and (discovered only recently,
and often called his most important work) The Method.
He developed the Stomachion puzzle (and solved a difficult
enumeration problem involving it); other famous gems
include The Cattle-Problem.
The Book of Lemmas contains various geometric gems
("the Salinon," "the Shoemaker's Knife", etc.) and is credited
to Archimedes by Thabit ibn Qurra but the attribution is disputed.

Archimedes discovered formulae for the volume and surface area
of a sphere, and may even have been first to notice and prove the
simple relationship between a circle's circumference and area.
For these reasons,
π is often called Archimedes' constant.
His approximation 223/71 < π < 22/7
was the best of his day.
(Apollonius soon surpassed it, but by using Archimedes' method.)
Archimedes' Equiarea Map Theorem asserts that a sphere and its enclosing
cylinder have equal surface area (as do the figures' truncations).
Archimedes also proved that the volume of that sphere is two-thirds the volume
of the cylinder.
He requested that a representation of such a sphere and
cylinder be inscribed on his tomb.

That Archimedes shared the attitude of later mathematicians like
Hardy and Brouwer is suggested by Plutarch's comment that Archimedes regarded
applied mathematics "as ignoble and sordid ... and did not deign
to [write about his mechanical inventions; instead]
he placed his whole ambition in those speculations the beauty and subtlety of
which are untainted by any admixture of the common needs of life."

Some of Archimedes' greatest writings (including
The Method and Floating Bodies) are preserved only on a
palimpsest rediscovered in 1906 and mostly deciphered only after 1998.
Ideas unique to that work are an anticipation of Riemann
integration, calculating
the volume of a cylindrical wedge (previously first attributed to Kepler);
along with Oresme and Galileo he was among the few to comment
on the "equinumerosity paradox" (the fact that are as many perfect
squares as integers).
Although Euler and Newton
may have been the most important mathematicians,
and Gauss, Weierstrass and Riemann the greatest theorem provers,
it is widely accepted that
Archimedes was the greatest genius who ever lived.
Yet, Hart omits him altogether from his list of Most Influential Persons:
Archimedes was simply too far ahead of his time to have great historical
significance.
(Some think the Scientific Revolution would have begun sooner
had The Method been discovered four or five centuries earlier.
You can
read a 1912 translation of parts of The Method on-line.)

Eratosthenes was one of the greatest polymaths; he is called
the Father of Geography, was Chief Librarian at Alexandria, was a
poet, music theorist,
mechanical engineer (anticipating laws of elasticity, etc.),
astronomer (he is credited as first to measure the circumference
of the Earth), and an outstanding mathematician.
He is famous for his prime number Sieve, but more impressive was his
work on the cube-doubling problem which he related to the design
of siege weapons (catapults) where a cube-root calculation is needed.

Eratosthenes had the nickname Beta; he was a master of several
fields, but was only second-best of his time.
His better was also his good friend:
Archimedes of Syracuse dedicated The Method to Eratosthenes.

Apollonius Pergaeus, called "The Great Geometer,"
is sometimes considered the second greatest
of ancient Greek mathematicians. (Euclid, Eudoxus and Archytas
are other candidates for this honor.)
His writings on conic sections have been studied until
modern times;
he developed methods for normals and curvature.
(He is often credited with inventing the names for parabola,
hyperbola and ellipse; but these shapes were previously described
by Menaechmus, and their names may also predate Apollonius.)
Although astronomers eventually concluded it was not physically correct,
Apollonius developed the "epicycle and deferent" model of planetary orbits,
and proved important theorems in this area.
He deliberately emphasized the beauty of pure, rather than
applied, mathematics, saying his theorems were
"worthy of acceptance for the sake of the demonstrations themselves."
The following
generalization of the Pythagorean Theorem, where M is the midpoint of BC,
is called Apollonius' Theorem:
AB 2 + AC 2 =
2(AM 2 + BM 2).

Many of his works have survived only in a fragmentary form,
and the proofs were completely lost.
Most famous was the Problem of Apollonius,
which is to find a circle tangent to three objects, with the
objects being points, lines, or circles, in any combination.
Constructing the eight circles each tangent to three other circles
is especially challenging, but just finding the two circles
containing two given points and tangent to a given line is
a serious challenge.
Vieta was renowned for discovering methods for all ten
cases of this Problem.
Other great mathematicians who have enjoyed reconstructing
Apollonius' lost theorems
include Fermat, Pascal, Newton, Euler, Poncelet and Gauss.

In evaluating the genius of the ancient Greeks,
it is well to remember that their achievements were made
without the convenience of modern notation.
It is clear from his writing that Apollonius almost developed
the analytic geometry of Descartes, but
failed due to the lack of such elementary concepts as negative numbers.
Leibniz wrote "He who understands Archimedes and Apollonius will admire less
the achievements of the foremost men of later times."

Chinese mathematicians excelled for thousands of years,
and were first to discover various algebraic and geometric principles.
There is some evidence that Chinese writings influenced India and
the Islamic Empire, and thus, indirectly, Europe.
Although there were great Chinese mathematicians a thousand years before
the Han Dynasty (as evidenced by the ancient Zhoubi Suanjing),
and innovations continued for centuries after Han,
the textbook Nine Chapters on the Mathematical Art
has special importance.
Nine Chapters (known in Chinese as Jiu Zhang Suan Shu
or Chiu Chang Suan Shu)
was apparently written during the early Han Dynasty (about 165 BC)
by Chang Tshang (also spelled Zhang Cang).

Many of the mathematical concepts of the early Greeks were
discovered independently in early China.
Chang's book gives methods of arithmetic (including cube roots)
and algebra,
uses the decimal system (though zero was represented as just a space,
rather than a discrete symbol),
proves the Pythagorean Theorem,
and includes a clever geometric proof that the perimeter of
a right triangle times the radius of its inscribing
circle equals the area of its circumscribing rectangle.
(Some of this may have been added after the time of Chang;
some additions attributed to Liu Hui are mentioned in his mini-bio;
other famous contributors are Jing Fang and Zhang Heng.)

Nine Chapters was probably based on earlier books,
lost during the great book burning of 212 BC, and
Chang himself may have been a lord who commissioned others to
prepare the book.
Moreover, important revisions and commentaries were added
after Chang, notably by Liu Hui (ca 220-280).
Although Liu Hui mentions Chang's skill, it isn't clear
Chang had the mathematical genius to qualify for this list,
but he would still be a strong candidate due to his book's
immense historical importance:
It was the dominant Chinese mathematical text for centuries,
and had great influence throughout the Far East.
After Chang,
Chinese mathematics continued to flourish, discovering
trigonometry, matrix methods, the Binomial Theorem, etc.
Some of the teachings made their way to India,
and from there to the Islamic world and Europe.
There is some evidence that the Hindus borrowed the
decimal system itself from books like Nine Chapters.

No one person can be credited with the invention of
the decimal system, but key roles were played by early Chinese
(Chang Tshang and Liu Hui),
Brahmagupta (and earlier Hindus including Aryabhata),
and Leonardo Fibonacci.
(After Fibonacci, Europe still did not embrace the decimal system
until the works of Vieta, Stevin, and Napier.)

Ptolemy may be the most famous astronomer
before Copernicus, but he borrowed heavily from Hipparchus,
who should thus be considered (along with Galileo and Edwin Hubble)
to be one of the three greatest astronomers ever.
Careful study of the errors in the catalogs
of Ptolemy and Hipparchus reveal both that Ptolemy
borrowed his data from Hipparchus, and that Hipparchus
used principles of spherical trig to simplify his work.
Classical Hindu astronomers, including the 6th-century genius
Aryabhata, borrow much from Ptolemy and Hipparchus.

Hipparchus is called the "Father of Trigonometry"; he
developed spherical trigonometry,
produced trig tables, and more.
He produced at least fourteen texts of physics and mathematics
nearly all of which have been lost, but which seem to
have had great teachings, including
much of Newton's Laws of Motion.
In one obscure surviving work he demonstrates familiarity
with the combinatorial enumeration method now called Schröder's Numbers.
He invented the circle-conformal stereographic
and orthographic map projections which carry his name.
As an astronomer, Hipparchus is credited with the discovery
of equinox precession, length of the year, thorough
star catalogs, and invention
of the armillary sphere and perhaps the astrolabe.
He had great historical influence in Europe, India and
Persia, at least if credited also with Ptolemy's influence.
(Hipparchus himself was influenced by Babylonian astronomers.)
Hipparchus' work implies a better approximation to π
than that of Apollonius, perhaps it was π ≈ 377/120 as
Ptolemy used.

The Antikythera mechanism is an astronomical clock
considered amazing for its time.
It may have been built about the time of Hipparchus'
death, but lost after a few decades
(remaining at the bottom of the sea for 2000 years).
The mechanism implemented the complex orbits which Hipparchus had developed
to explain irregular planetary motions;
it's not unlikely the great genius helped design this intricate
analog computer, which may have been built in Rhodes
where Hipparchus spent his final decades.
(Recent studies suggest that the mechanism was designed
in Archimedes' time, and that therefore that genius
might have been the designer.)

Menelaus wrote several books on geometry and
trigonometry, mostly lost except for his works on solid geometry.
His work was cited by Ptolemy, Pappus, and Thabit;
especially the Theorem of Menelaus itself which is a
fundamental and difficult theorem very useful in projective geometry.
He also contributed much to spherical trigonometry.
Disdaining indirect proofs (anticipating later-day constructivists)
Menelaus found new, more fruitful proofs for several of Euclid's results.

Ptolemy, the Librarian of Alexandria,
was one of the most famous of ancient Greek scientists.
Among his mathematical results, most famous may be Ptolemy's Theorem
(AC·BD = AB·CD + BC·AD
if and only if ABCD is a cyclic quadrilateral).
This theorem has many useful corollaries; it was frequently applied
in Copernicus' work.
Ptolemy also wrote on trigonometry, optics, geography, map projections,
and astrology; but is most famous for his astronomy,
where he perfected the geocentric model of planetary motions.
For this work, Cardano included Ptolemy on his List of 12 Greatest Geniuses,
but removed him from the list after learning of Copernicus' discovery.
Interestingly, Ptolemy wrote that the fixed point in a model of planetary
motion was arbitrary, but rejected the Earth spinning on its axis
since he thought this would lead to powerful winds.
Ptolemy discussed and tabulated the 'equation of time,' documenting the
irregular apparent motion of the Sun. (It took fifteen centuries
before this irregularity was correctly attributed to Earth's elliptical orbit.)

Geocentrism vs. Heliocentrism

The mystery of celestial motions directed scientific inquiry for thousands
of years.
With the notable exception of the Pythagorean Philolaus of Croton,
thinkers generally assumed that the Earth was the center of the universe,
but this made it very difficult to explain the orbits of the other planets.
This problem had been considered by Eudoxus, Apollonius, and Hipparchus,
who developed a very complicated geocentric model involving
concentric spheres and epicyles.
Ptolemy perfected (or, rather, complicated) this model even further,
introducing 'equants' to further fine-tune the orbital speeds;
this model was the standard for 14 centuries.
While some Greeks, notably Aristarchus and Seleucus of Seleucia
(and perhaps also Heraclides of Pontus or ancient Egyptians),
proposed heliocentric models,
these were rejected because there was no parallax among stars.
(Aristarchus guessed that the stars were at an almost unimaginable
distance, explaining the lack of parallax.
Aristarchus would be almost unknown except that
Archimedes mentions, and assumes, Aristarchus' heliocentrism in
The Sand Reckoner.
I suspect that Archimedes accepted heliocentrism, but thought saying
so openly would distract from his work.
Hipparchus was another ancient Greek who considered heliocentrism but,
because he never guessed
that orbits were ellipses rather than cascaded circles, was unable to come
up with a heliocentric model that fit his data.)
Aryabhata, Alhazen, Alberuni, Omar Khayyám, (perhaps some other
Islamic mathematicians like al-Tusi), and Regiomontanus are
other great pre-Copernican mathematicians who may have accepted
the possibility of heliocentrism.

The great skill demonstrated by Ptolemy and his predecessors in
developing their complex geocentric cosmology
may have set back science since in fact
the Earth rotates around the Sun.
The geocentric models couldn't
explain the observed changes in the brightness of Mars or Venus,
but it was the phases of Venus, discovered by Galileo after the invention
of the telescope, that finally led to general acceptance of heliocentrism.
(Ptolemy's model predicted phases, but timed quite differently from
Galileo's observations.)

Since the planets move without friction, their motions offer a pure
view of the Laws of Motion; this is one reason that the heliocentric
breakthroughs of Copernicus, Kepler and Newton triggered the advances
in mathematical physics which led to the Scientific Revolution.
Heliocentrism offered an even more key understanding that lead to
massive change in scientific thought.
For Ptolemy and other geocentrists, the "fixed" stars
were just lights on a sphere rotating around the earth, but after
the Copernican Revolution the fixed stars were understood to be
immensely far away; this made it possible to imagine that they were
themselves suns, perhaps with planets of their own. (Nicole Oresme
and Nicholas of Cusa were pre-Copernican thinkers who wrote on both
the geocentric question and the possibility of other worlds.)
The Copernican perspective led
Giordano Bruno and Galileo to posit a single common set of
physical laws which ruled both on Earth and in the Heavens.
(It was this, rather than just the happenstance of planetary orbits,
that eventually most outraged the Roman Church....
And we're getting ahead of our
story: Copernicus, Bruno, Galileo and Kepler lived 14 centuries after Ptolemy.)

Liu Hui made major improvements
to Chang's influential textbook Nine Chapters,
making him among the most important of Chinese mathematicians ever.
(He seems to have been a much better mathematician
than Chang, but just as Newton might have gotten nowhere without Kepler,
Vieta, Huygens, Fermat, Wallis, Cavalieri, etc., so Liu Hui might
have achieved little had Chang not preserved the ancient
Chinese learnings.)
Among Liu's achievements
are an emphasis on generalizations and proofs,
incorporation of negative numbers into arithmetic,
an early recognition of the notions of infinitesimals and limits,
the Gaussian elimination method of solving
simultaneous linear equations,
calculations of solid volumes (including the use of Cavalieri's Principle),
anticipation of Horner's Method,
and a new method to calculate square roots.
Like Archimedes, Liu discovered the formula for a circle's area;
however he failed to calculate a sphere's volume, writing
"Let us leave this problem to whoever can tell the truth."

Although it was almost child's-play for any of them,
Archimedes, Apollonius, and Hipparchus had all improved
precision of π's estimate.
It seems fitting that Liu Hui did join that select company of
record setters: He developed a recurrence formula for
regular polygons allowing arbitrarily-close approximations
for π.
He also devised an interpolation formula to simplify
that calculation; this yielded the "good-enough" value 3.1416,
which is still taught today in primary schools.
(Liu's successors in China included Zu Chongzhi, who did
determine sphere's volume, and whose approximation for π
held the accuracy record for nine centuries.)

Diophantus was one of the most influential
mathematicians of antiquity; he wrote several books on
arithmetic and algebra,
and explored number theory further than anyone earlier.
He advanced a rudimentary arithmetic and algebraic notation, allowed
rational-number solutions to his problems rather than just integers,
and was aware of results like the Brahmagupta-Fibonacci Identity;
for these reasons he is often called the "Father of Algebra."
His work, however, may seem quite limited to a modern eye:
his methods were not generalized, he knew nothing
of negative numbers, and, though he often dealt with quadratic
equations, never seems to have commented on their second solution.
His notation, clumsy as it was, was used for many centuries.
(The shorthand x3
for "x cubed" was not invented until Descartes.)

Very little is known about Diophantus (he might even have come from
Babylonia, whose algebraic ideas he borrowed).
Many of his works have been lost, including proofs for lemmas
cited in the surviving work, some of which are so
difficult it would almost stagger the imagination to
believe Diophantus really had proofs.
Among these are Fermat's conjecture (Lagrange's theorem)
that every integer is the sum of four squares, and the
following:
"Given any positive rationals a, b with a>b,
there exist positive rationals c, d such that
a3-b3 = c3+d3."
(This latter "lemma" was investigated by Vieta and Fermat and
finally solved, with some difficulty, in the 19th century.
It seems unlikely that Diophantus actually had proofs for such "lemmas.")

Pappus, along with Diophantus, may have been
one of the two greatest Western mathematicians
during the 13 centuries that separated Hipparchus and Fibonacci.
He wrote about arithmetic methods, plane and solid geometry,
the axiomatic method, celestial motions and mechanics.
In addition to his own original research, his texts are
noteworthy for preserving works of earlier mathematicians
that would otherwise have been lost.

Pappus' best and most original result, and the one which gave
him most pride, may be the Pappus Centroid theorems
(fundamental, difficult and powerful theorems of solid geometry
later rediscovered by Paul Guldin).
His other ingenious geometric theorems
include Desargues' Homology Theorem (which Pappus attributes
to Euclid), an early form of Pascal's Hexagram Theorem,
called Pappus' Hexagon Theorem and related to a fundamental
theorem: Two projective pencils
can always be brought into a perspective position.
For these theorems, Pappus is sometimes called
the "Father of Projective Geometry."
Pappus also demonstrated how to perform angle trisection and
cube doubling if one can use mechanical curves like
a conchoid or hyperbola.
He stated (but didn't prove) the Isoperimetric Theorem, also
writing "Bees know this fact which is useful to them,
that the hexagon ... will hold more honey
for the same material than [a square or triangle]."
(That a honeycomb partition minimizes material for an equal-area
partitioning was finally proved in 1999 by Thomas Hales, who also
proved the related Kepler Conjecture.)
Pappus stated, but did not fully solve, the Problem of Pappus
which, given an arbitrary collection of lines in the plane, asks for
the locus of points whose distances to the lines have
a certain relationship.
This problem was a major inspiration for Descartes and was finally
fully solved by Newton.

For preserving the teachings of Euclid and Apollonius,
as well as his own theorems of geometry, Pappus certainly
belongs on a list of great ancient mathematicians.
But these teachings lay dormant during Europe's Dark Ages, diminishing
Pappus' historical significance.

Alexander the Great spread Greek culture to Egypt and much of the Orient;
thus even Hindu mathematics may owe something to the Greeks.
Greece was eventually absorbed into the Roman Empire
(with Archimedes himself famously killed by a Roman soldier).
Rome did not pursue pure science as Greece had (as we've seen,
the important mathematicians of the Roman era
were based in the Hellenic East)
and eventually Europe fell into a Dark Age.
The Greek emphasis on pure mathematics and proofs was
key to the future of mathematics, but they were missing
an even more important catalyst: a decimal place-value system
based on zero and nine other symbols.

Laplace called the decimal system "a profound and important idea
[given by India] which
appears so simple to us now that we ignore its true merit ... in
the first rank of useful inventions [but] it escaped the
genius of Archimedes and Apollonius."
But even after Fibonacci introduced the system to Europe, it was another
400 years before it came into common use.

Ancient Greeks, by the way, did not use the
unwieldy Roman numerals, but rather used 27 symbols, denoting
1 to 9, 10 to 90, and 100 to 900.
Unlike our system, with ten digits separate from the alphabet,
the 27 Greek number symbols were the same as their
alphabet's letters; this might have hindered
the development of "syncopated" notation.
The most ancient Hindu records did not use the
ten digits of Aryabhata, but rather a system
similar to that of the ancient Greeks, suggesting that
China, and not India, may indeed be the "ultimate" source of the modern
decimal system.

The Chinese used a form of decimal abacus as early as 3000 BC;
if it doesn't qualify, by itself, as a "decimal system" then
pictorial depictions of its numbers would.
Yet for thousands of years after its abacus, China had no zero symbol
other than plain space; and apparently didn't have one until after
the Hindus. Ancient Persians and Mayans did have place-value
notation with zero symbols, but neither qualify as inventing
a base-10 decimal system: Persia used the base-60 Babylonian system;
Mayans used base-20.
(Another difference is that the Hindus had nine distinct digit symbols
to go with their zero, while earlier place-value systems
built up from just two symbols: 1 and either 5 or 10.)
The Old Kingdom Egyptians did use a base-ten system, but it was
similar to that of Greece and Vedic India:
1, 10, 100 were depicted as separate symbols.

Conclusion: The decimal place-value system with zero symbol seems
to be an obvious invention that in fact was very hard to invent.
If you insist on a single winner then India might be it.
But China, Babylonia, Persia and even the Mayans deserve Honorable Mention!

Indian mathematicians excelled for thousands of years,
and eventually even developed advanced techniques like Taylor series
before Europeans did, but they are
denied credit because of Western ascendancy.
Among the Hindu mathematicians, Aryabhata (called Arjehir by Arabs)
may be most famous.

While Europe was in its early "Dark Age,"
Aryabhata advanced arithmetic, algebra,
elementary analysis, and especially
plane and spherical trigonometry, using the decimal system.
Aryabhata is sometimes called the "Father of Algebra"
instead of al-Khowârizmi (who himself cites the work of Aryabhata).
His most famous accomplishment in mathematics
was the Aryabhata Algorithm (connected
to continued fractions) for solving Diophantine equations.
Aryabhata made several important discoveries in astronomy,
e.g. the nature of moonlight, and concept of sidereal year;
his estimate of the Earth's circumference was more accurate than
any achieved in ancient Greece.
He was among the very few ancient scholars who realized the Earth
rotated daily on an axis; claims that he also espoused heliocentric
orbits are controversial, but may be confirmed by the writings of al-Biruni.
Aryabhata is said to have introduced the constant e.
He used π ≈ 3.1416;
it is unclear whether he discovered this independently
or borrowed it from Liu Hui of China.
Although it was first discovered by Nicomachus three centuries earlier,
Aryabhata is famous for the identity
Σ (k3) = (Σ k)2

Some of Aryabhata's achievements, e.g. an excellent approximation
to the sine function, are known only from the writings of
Bhaskara I, who wrote: "Aryabhata is
the master who, after reaching the furthest shores and plumbing the inmost
depths of the sea of ultimate knowledge of mathematics, kinematics
and spherics, handed over the three sciences to the learned world."

No one person gets unique credit for the invention of the
decimal system but Brahmagupta's textbook Brahmasphutasiddhanta
was very influential, and is sometimes considered
the first textbook "to treat zero as a number in its own right."
It also treated negative numbers.
(Others claim these were first seen 800 years earlier in
Chang Tshang's Chinese text and were implicit in
what survives of earlier Hindu works, but Brahmagupta's
text discussed them lucidly.)
Along with Diophantus, Brahmagupta was also among the first to express
equations with symbols rather than words.

Brahmagupta Bhillamalacarya (`The Teacher from Bhillamala')
made great advances in arithmetic,
algebra, numeric analysis, and geometry.
Several theorems bear his name, including
the formula for the area of a cyclic quadrilateral:
16 A2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)Another famous Brahmagupta theorem dealing with
such quadrilaterals can be phrased
"In a circle, if the chords AB and CD are perpendicular
and intersect at E, then the line from E which bisects AC
will be perpendicular to BD."
He also began the study of rational quadrilaterals which Kummer would
eventually complete.
Proving Brahmagupta's theorems are good challenges even today.

In addition to his famous writings on practical mathematics
and his ingenious theorems of geometry,
Brahmagupta solved the general quadratic equation,
and worked on number theory problems.
He was first to find a general solution to the simplest Diophantine
form.
His work on Pell's equations has been called "brilliant"
and "marvelous."
He proved the Brahmagupta-Fibonacci Identity
(the set of sums of two squares is closed under multiplication).
He applied mathematics to astronomy, predicting eclipses, etc.

The astronomer Bháskara I, who takes the suffix "I"
to distinguish him from
the more famous Bháskara who lived five centuries later,
made key advances to the positional decimal number notation,
and was the first known to use the zero symbol.
He preserved some of the teachings of Aryabhata which would
otherwise have been lost; these include a famous
formula giving an excellent approximation to the
sin function, as well as, probably, the zero symbol itself.

Among other original contributions to mathematics,
Bháskara I was first to state Wilson's Theorem (which should
perhaps be called Bháskara's Conjecture):
(n-1)! ≡ -1 (mod n) if and only if n is prime
Bháskara's Conjecture was rediscovered by Alhazen, Fibonacci,
Leibniz and Wilson. The "only if" is easy but the difficult "if"
part was finally proved by Lagrange in 1771.
Since Lagrange has so many other Theorems named
after him, Bháskara's Conjecture is always called "Wilson's Theorem."

Al-Khowârizmi (aka Mahomet ibn Moses) was a Persian
who worked as a mathematician, astronomer and geographer
early in the Golden Age of Islamic science.
He introduced the Hindu decimal system to the Islamic world and Europe;
invented the horary quadrant; improved the sundial;
developed trigonometry tables; and improved on Ptolemy's
astronomy and geography.
He wrote the book
Al-Jabr, which demonstrated simple algebra and geometry,
and several other influential books.
Unlike Diophantus' work, which dealt in specific examples,
Al-Khowârizmi was the first algebra text to present general methods;
he is often called the "Father of Algebra."
(Diophantus did, however, use superior "syncopated" notation.)
The word algorithm is borrowed from Al-Khowârizmi's name,
and algebra is taken from the name of his book.
He also coined the word cipher, which became English zero
(although this was just a translation from the Sanskrit word
for zero introduced by Aryabhata).
He was an essential pioneer for Islamic science,
and for the many Arab and Persian mathematicians who followed;
and hence also for Europe's eventual Renaissance which was heavily dependent
on Islamic teachings.
Al-Khowârizmi's texts on algebra and decimal arithmetic are
considered to be among the most influential writings ever.

Al-Kindi (called Alkindus or Achindus in the West)
wrote on diverse philosophical subjects, physics, optics,
astronomy, music, psychology, medicine, chemistry, and more.
He invented pharmaceutical methods, perfumes, and distilling of alcohol.
In mathematics, he popularized the use of the decimal system,
developed spherical geometry, wrote on many other topics and was
a pioneer of cryptography (code-breaking).
(His work with code-breakig also made him a pioneer in
basic concepts of probability.)
(Al-Kindi, called The Arab Philosopher, can not be
considered among the greatest of mathematicians,
but was one of the most influential general
scientists between Aristotle and da Vinci.)
He appears on Cardano's List of 12 Greatest Geniuses.
(Al-Khowârizmi and Jabir ibn Aflah are the other
Islamic scientists on that list.)

Thabit produced important books in philosophy
(including perhaps the famous mystic work De Imaginibus),
medicine, mechanics, astronomy, and especially several mathematical fields:
analysis, non-Euclidean geometry, trigonometry, arithmetic,
number theory.
As well as being an original thinker, Thabit was a key
translator of ancient Greek writings; he translated
Archimedes' otherwise-lost Book of Lemmas and applied
one of its methods to construct a regular heptagon.
He developed an important new cosmology superior
to Ptolemy's (and which, though it was not heliocentric,
may have inspired Copernicus).
He was perhaps the first great mathematician to take the
important step of emphasizing
real numbers rather than either rational numbers or geometric sizes.
He worked in plane and spherical trigonometry,
and with cubic equations.
He was an early practitioner of calculus and seems to
have been first to take the integral of √x.
Like Archimedes, he was
able to calculate the area of an ellipse,
and to calculate the volume of a paraboloid.
He produced an elegant generalization of the
Pythagorean Theorem:
AC 2 + BC 2 = AB (AR + BS)(Here the triangle ABC is not a right triangle, but R and S are located
on AB to give the equal angles ACB = ARC = BSC.)
Thabit also worked in number theory where he is especially famous
for his theorem about amicable numbers.
(Thabit ibn Qurra's Theorem was rediscovered by Fermat and Descartes, and later
generalized by Euler.)
While many of his discoveries in geometry, plane and spherical trigonometry,
and analysis (parabola quadrature, trigonometric law, principle
of lever) duplicated work by Archimedes and Pappus, Thabit's
list of novel achievements is impressive.
Among the several great and famous Baghdad geometers,
Thabit may have had the greatest genius.

Ibn Sinan, grandson of Thabit ibn Qurra, was one of the greatest
Islamic mathematicians and might have surpassed his famous grandfather
had he not died at a young age.
He was an early pioneer of analytic geometry,
advancing the theory of integration, applying algebra to synthetic geometry,
and writing on the construction of conic sections.
He produced a new proof of Archimedes' famous formula for the area of
a parabolic section.
He worked on the theory of area-preserving transformations, with
applications to map-making.
He also advanced astronomical theory, and wrote a treatise on sundials.

Al-Hassan ibn al-Haytham (Alhazen)
made contributions to math, optics, and astronomy
which eventually influenced Roger Bacon, Regiomontanus, da Vinci, Copernicus,
Kepler, Galileo, Huygens, Descartes and Wallis,
thus affecting Europe's Scientific Revolution.
He's been called the best scientist of the Middle Ages;
his Book of Optics has been called the most important
physics text prior to Newton;
his writings in physics anticipate the Principle of Least Action,
Newton's First Law of Motion, and the notion that white light
is composed of the color spectrum.
(Like Newton, he favored a particle theory of light over the
wave theory of Aristotle.)
His other achievements in optics include improved lens design,
an analysis of the camera obscura, Snell's Law,
an early explanation for the rainbow,
a correct deduction from refraction of atmospheric thickness,
and experiments on visual perception.
He studied optical illusions and was first to explain
psychologically why the Moon appears to be larger when near
the horizon.
He also did work in human anatomy and medicine.
(In a famous leap of over-confidence he claimed he could
control the Nile River; when the Caliph ordered him to do so,
he then had to feign madness!)
Alhazen has been called the "Father of Modern Optics,"
the "Founder of Experimental Psychology" (mainly for his work with
optical illusions), and,
because he emphasized hypotheses and experiments,
"The First Scientist."

In number theory, Alhazen worked with perfect numbers, Mersenne primes,
and the Chinese Remainder Theorem.
He stated Wilson's Theorem (which is sometimes called Al-Haytham's Theorem).
Alhazen introduced the Power Series Theorem
(later attributed to Jacob Bernoulli).
His best mathematical work was with plane and solid geometry,
especially conic sections; he calculated the areas of lunes, volumes
of paraboloids, and constructed a heptagon using
intersecting parabolas.
He solved Alhazen's Billiard Problem (originally posed as a problem
in mirror design), a difficult construction which continued to intrigue
several great mathematicians including Huygens.
To solve it, Alhazen needed to
anticipate Descartes' analytic geometry,
anticipate Bézout's Theorem,
tackle quartic equations
and develop a rudimentary integral calculus.
Alhazen's attempts to prove the Parallel Postulate make him (along
with Thabit ibn Qurra) one of the earliest
mathematicians to investigate non-Euclidean geometry.

Al-Biruni (Alberuni) was an extremely outstanding scholar,
far ahead of his time, sometimes shown with
Alkindus and Alhazen as one of the greatest Islamic polymaths,
and sometimes compared to Leonardo da Vinci.
He is less famous in part because he lived in a remote part of the
Islamic empire.
He was a great linguist; studied the original works of Greeks and Hindus;
is famous for debates with his contemporary Avicenna;
studied history, biology, mineralogy,
philosophy, sociology, medicine and more;
is called the Father of Geodesy and
the Father of Arabic Pharmacy;
and was one of the greatest astronomers.
He was an early advocate of the Scientific Method.
He was also noted for his poetry.
He invented (but didn't build) a geared-astrolabe clock,
and worked with springs and hydrostatics.
He wrote prodigiously on all scientific topics (his writings are estimated
to total 13,000 folios); he was especially noted for
his comprehensive encyclopedia about India, and Shadows,
which starts from notions about shadows but develops much astronomy
and mathematics.
He anticipated future advances
including Darwin's natural selection,
Newton's Second Law, the immutability of elements,
the nature of the Milky Way, and much modern geology.
Among several novel achievements in astronomy, he used
observations of lunar eclipse to deduce relative longitude,
estimated Earth's radius most accurately,
believed the Earth rotated on its
axis and may have accepted heliocentrism as a possibility.
In mathematics, he was first to apply the Law of Sines
to astronomy, geodesy, and cartography;
anticipated the notion of polar coordinates;
invented the azimuthal equidistant map projection in common use today,
as well as a polyconic method
now called the Nicolosi Globular Projection;
found trigonometric solutions to polynomial equations;
did geometric constructions including angle trisection;
and wrote on arithmetic, algebra, and combinatorics
as well as plane and spherical trigonometry and geometry.
(Al-Biruni's contemporary Avicenna was not particularly a mathematician
but deserves mention as an advancing scientist, as does Avicenna's disciple
Abu'l-Barakat al-Baghdada, who lived about a century later.)

Al-Biruni has left us what seems to be the oldest surviving mention
of the Broken Chord Theorem (if M is the midpoint of circular arc ABMC,
and T the midpoint of "broken chord" ABC, then MT is perpendicular to BC).
Although he himself attributed the theorem to Archimedes,
Al-Biruni provided several novel proofs for, and useful corollaries of,
this famous geometric gem.
While Al-Biruni may lack the influence and mathematical brilliance to
qualify for the Top 100, he deserves recognition as one of the
greatest applied mathematicians before the modern era.

Omar Khayyám (aka Ghiyas od-Din Abol-Fath Omar ibn Ebrahim
Khayyam Neyshaburi) was one of the greatest Islamic mathematicians.
He did clever work with geometry, developing
an alternate to Euclid's Parallel Postulate and then deriving the
parallel result using theorems
based on the Khayyam-Saccheri quadrilateral.
He derived solutions to cubic equations using the intersection
of conic sections with circles.
Remarkably, he stated that the cubic solution could not be achieved
with straightedge and compass, a fact that wouldn't
be proved until the 19th century.
Khayyám did even more important work in algebra, writing
an influential textbook, and developing new solutions for
various higher-degree equations.
He may have been first to develop Pascal's Triangle (which is still
called Khayyám's Triangle in Persia), along with the
essential Binomial Theorem (Al-Khayyám's Formula):
(x+y)n = n! Σ xkyn-k / k!(n-k)!

Khayyám was also an important astronomer; he measured the
year far more accurately than ever before,
improved the Persian calendar, built a famous star map,
and believed that the Earth rotates on its axis.
He was a polymath: in addition to being a philosopher of far-ranging scope,
he also wrote treatises on music, mechanics and natural science.
He was noted for deriving his theories from science rather than religion.
Despite his great achievements in algebra, geometry,
astronomy, and philosophy, today Omar al-Khayyám is most
famous for his rich poetry (The Rubaiyat of Omar Khayyám).

Bháscara II (also called Bhaskaracharya)
may have been the greatest of the Hindu
mathematicians. He made achievements in several fields
of mathematics including some Europe wouldn't learn until the time of Euler.
His textbooks dealt with many matters, including solid geometry, combinations,
and advanced arithmetic methods.
He was also an astronomer.
(It is sometimes claimed that his
equations for planetary motions anticipated the Laws of Motion
discovered by Kepler and Newton, but this claim is doubtful.)
In algebra, he solved various equations including 2nd-order
Diophantine, quartic, Brouncker's and Pell's equations.
His Chakravala method, an early application of mathematical induction
to solve 2nd-order equations, has been called "the finest thing
achieved in the theory of numbers before Lagrange"
(although a similar statement was made about one of Fibonacci's theorems).
(Earlier Hindus, including Brahmagupta, contributed to this method.)
In several ways he anticipated calculus: he used Rolle's Theorem;
he may have been first to use the fact
that dsin x = cos x · dx;
and he once wrote that multiplication by 0/0 could be "useful
in astronomy."
In trigonometry, which he valued for its own beauty as well
as practical applications, he developed spherical trig and was first to
present the identity
sin a+b
= sin a · cos b
+ sin b · cos a

Bháscara's achievements came centuries before similar
discoveries in Europe.
It is an open riddle of history whether any of Bháscara's
teachings trickled into Europe in time to influence its
Scientific Renaissance.

Leonardo (known today as Fibonacci)
introduced the decimal system and other new methods of arithmetic to Europe,
and relayed the mathematics of the Hindus, Persians, and Arabs.
Others, especially Gherard of Cremona, had translated Islamic
mathematics, e.g. the works of al-Khowârizmi, into Latin,
but Leonardo was the influential teacher.
(Two centuries earlier, the mathematician-Pope, Gerbert of Aurillac,
had tried unsuccessfully to introduce the decimal system to Europe.)
Leonardo also re-introduced older Greek ideas like
Mersenne numbers and Diophantine equations.
His writings cover a very broad range including
new theorems of geometry,
methods to construct and convert Egyptian fractions
(which were still in wide use),
irrational numbers,
the Chinese Remainder Theorem,
theorems about Pythagorean triplets,
and the series 1, 1, 2, 3, 5, 8, 13, ....
which is now linked with the name Fibonacci.
In addition to his great historic importance and fame
(he was a favorite of Emperor Frederick II),
Leonardo `Fibonacci' is called "the greatest number theorist
between Diophantus and Fermat" and "the most talented
mathematician of the Middle Ages."

Leonardo is most famous for his book Liber Abaci, but
his Liber Quadratorum provides the best demonstration of his skill.
He defined congruums and proved theorems about them,
including a theorem establishing the conditions for three square numbers
to be in consecutive arithmetic series; this has been called
the finest work in number theory prior to Fermat
(although a similar statement was made about one of Bhaskara II's theorems).
Although often overlooked, this work includes a proof
of the n = 4 case of Fermat's Last Theorem.
(Leonardo's proof of FLT4 is widely ignored or considered incomplete.
I'm preparing a page to consider that question.
Al-Farisi was another ancient mathematician who noted FLT4, although
attempting no proof.)
Another of Leonardo's noteworthy achievements was proving that
the roots of a certain cubic equation could not have any of the
constructible forms Euclid had outlined in Book 10 of his Elements.
He also wrote on, but didn't prove, Wilson's Theorem.

Leonardo provided Europe with the decimal system,
algebra and the 'lattice' method of multiplication, all
far superior to the methods then in use.
He introduced notation like 3/5; his clever extension
of this for quantities like 5 yards, 2 feet, and 3 inches
is more efficient than today's notation.
It seems hard to believe but before the decimal
system, mathematicians had no notation for zero.
Referring to this system,
Gauss was later to exclaim "To what heights would science
now be raised if Archimedes had made that discovery!"

Some histories describe him as bringing Islamic mathematics
to Europe, but in Fibonacci's own preface to Liber Abaci,
he specifically credits the Hindus:

... as a consequence of marvelous instruction in the art,
to the nine digits of the Hindus, the knowledge of the art
very much appealed to me before all others, and for it I
realized that all its aspects were studied in Egypt, Syria,
Greece, Sicily, and Provence, with their varying methods;
... But all this even, and the algorism, as well as the art
of Pythagoras, I considered as almost a mistake in respect
to the method of the Hindus. Therefore,
embracing more stringently that method of the Hindus, and
taking stricter pains in its study, while adding certain
things from my own understanding and inserting also certain
things from the niceties of Euclid's geometric art, I have
striven to compose this book in its entirety as understandably
as I could, ...

Had the Scientific Renaissance begun in the Islamic Empire,
someone like al-Khowârizmi would have greater historic
significance than Fibonacci,
but the Renaissance did happen in Europe.
Liber Abaci's summary of the
decimal system has been called "the most important sentence
ever written."
Even granting this to be an exaggeration, there is no doubt
that the Scientific Revolution owes a huge debt to
Leonardo `Fibonacci' Pisano.

Al-Tusi was one of the greatest Islamic polymaths, working
in theology, ethics, logic, astronomy, and other fields of science.
He was a famous scholar and prolific writer,
describing evolution of species, stating that the Milky Way was
composed of stars,
and mentioning conservation of mass in his writings on chemistry.
He made a wide range of contributions to astronomy, and (along
with Omar Khayyám) was one of the
most significant astronomers between Ptolemy and Copernicus.
He improved on the Ptolemaic model of planetary orbits, and even wrote
about (though rejecting) the possibility of heliocentrism.

Tusi is most famous for his mathematics.
He advanced algebra, arithmetic, geometry, trigonometry, and even foundations,
working with real numbers and lengths of curves.
For his texts and theorems, he may be called the "Father of Trigonometry;"
he was first to properly state and prove several theorems
of planar and spherical trigonometry including the Law of Sines,
and the (spherical) Law of Tangents.
He wrote important
commentaries on works of earlier Greek and Islamic mathematicians;
he attempted to prove Euclid's Parallel Postulate.
Tusi's writings influenced European mathematicians including Wallis;
his revisions of the Ptolemaic model led him to the Tusi-couple,
a special case of trochoids usually called Copernicus' Theorem,
though historians have concluded Copernicus
discovered this theorem by reading Tusi.

There were several important Chinese mathematicians
in the 13th century, of whom Qin Jiushao (Ch'in Chiu-Shao) may have had
particularly outstanding breadth and genius.
Qin's textbook discusses various algebraic procedures,
includes word problems requiring quartic or quintic equations,
explains a version of Horner's Method for finding solutions
to such equations,
includes Heron's Formula for a triangle's area, and
introduces the zero symbol and decimal fractions.
Qin's work on the Chinese Remainder Theorem was very impressive,
finding solutions in cases which later stumped Euler.

Other great Chinese mathematicians of that era are
Li Zhi, Yang Hui (Pascal's Triangle is still
called Yang Hui's Triangle in China), and Zhu Shiejie.
Their teachings did not make their way to Europe, but were read by the
Japanese mathematician Seki, and possibly by Islamic
mathematicians like Al-Kashi.
Although Qin was a soldier and governor noted for corruption, with
mathematics just a hobby, I've chosen him to represent this
group because of the key advances which appear first in his writings.

Zhu Shiejie (Chu Shih-Chieh) was more famous and influential
than Qin; historian George Sarton called him
"one of the greatest mathematicians ... of all time."
His book Jade Mirror of the Four Unknowns studied
multivariate polynomials and is considered
the best mathematics in ancient China and describes methods not
rediscovered for centuries; for example
Zhu anticipated the Sylvester matrix method for solving simultaneous
polynomial equations.

Al-Farisi was a student of al-Shirazi who in turn was
a student of al-Tusi. He and al-Shirazi are especially noted for the
first correct explanation of the rainbow.
Al-Farisi made several other corrections in his comprehensive
commentary on Alhazen's textbook on optics.

Al-Farisi made several contributions to number theory. He improved
Thabit's theorem about amicable numbers, made important new observations
about the binomial coefficients, and noted the N=4 case of Fermat's Last
Theorem.
In addition to his work with amicable numbers, he is
especially noted for his improved proof of
Euclid's Fundamental Theorem of Arithmetic.

Gersonides (aka Leo de Bagnols, aka RaLBaG) was a Jewish scholar of
great renown, preferring science and reason over religious orthodoxy.
He wrote important commentaries on Aristotle, Euclid, the Talmud,
and the Bible; he is most famous for his
book MilHamot Adonai ("The Wars of the Lord") which
touches on many theological questions.
He was likely the most talented scientist of his time:
he invented the "Jacob's Staff" which became an important navigation
tool; described the principles of the camera obscura; etc.
In mathematics, Gersonides wrote texts on trigonometry,
calculation of cube roots, rules of arithmetic, etc.;
and gave rigorous derivations of rules of combinatorics.
He was first to make explicit use of mathematical induction.
At that time, "harmonic numbers" referred to integers with only 2 and
3 as prime factors; Gersonides solved a problem of music theory with
an ingenious proof that there were no consecutive harmonic numbers
larger than (8,9).
Levi ben Gerson published only in Hebrew so,
although some of his work was translated into
Latin during his lifetime, his influence was limited; much of his work was re-invented
three centuries later; and many histories of math overlook him altogether.

Gersonides was also an outstanding astronomer. He proved that
the fixed stars were at a huge distance, and found other
flaws in the Ptolemaic model.
But he specifically rejected heliocentrism, noteworthy since it
implies that heliocentrism was under consideration at the time.

Oresme was of lowly birth but excelled at school
(where he was taught by the famous Jean Buridan),
became a young professor, and soon personal chaplain to King Charles V.
The King commissioned him to translate the works of
Aristotle into French (with Oresme thus playing key roles in
the development of both French science and French language), and
rewarded him by making him a Bishop.
He wrote several books; was a renowned philosopher and natural scientist
(challenging several of Aristotle's ideas);
contributed to economics (e.g. anticipating Gresham's Law)
and to optics (he was first to posit curved refraction).
Although the Earth's annual orbit around the Sun was left to Copernicus,
Oresme was among the pre-Copernican thinkers to claim clearly that the
Earth spun daily on its axis.

In mathematics, Oresme observed that the integers were
equinumerous with the odd integers; was first to use fractional
(and even irrational) exponents;
introduced the symbol + for addition;
was first to write about general curvature;
and, most famously, first to prove the divergence of the harmonic series.
Oresme used a graphical diagram to demonstrate the Merton College Theorem
(a discovery related to Galileo's Law of Falling Bodies
made by Thomas Bradwardine, et al); it is said this was the
first abstract graph.
(Some believe that this effort inspired
Descartes' coordinate geometry and Galileo.)
Oresme was aware of Gersonides' work on harmonic numbers and was among
those who attempted to link music theory to the ratios of celestial
orbits, writing "the heavens are like a man who sings a melody and
at the same time dances, thus making music ... in song and in action."
Oresme's work was influential; with several discoveries ahead of
his time, Oresme deserves to be better known.

Madhava, also known as Irinjaatappilly Madhavan Namboodiri,
founded the important Kerala school of mathematics and astronomy.
If everything credited to him was his own work, he was a
truly great mathematician.
His analytic geometry preceded and surpassed Descartes',
and included differentiation and integration.
Madhava also did work with continued fractions, trigonometry,
and geometry.
He has been called the "Founder of Mathematical Analysis."
Madhava is most famous for his work with Taylor series,
discovering identities like
sin q = q - q3/3! +
q5/5! - ... ,
formulae for π, including the one attributed to Leibniz,
and the then-best known approximation
π ≈ 104348 / 33215.

Despite the accomplishments of the Kerala school, Madhava probably does not
deserve a place on our List. There were several other great mathematicians
who contributed to Kerala's achievements, some of which were made 150 years
after Madhava's death.
More importantly, the work was not propagated outside Kerala,
so had almost no effect on the development of mathematics.

Al-Kashi was among the greatest calculaters in the
ancient world; wrote important texts applying arithmetic and algebra
to problems in astronomy, mensuration and accounting;
and developed trig tables far more accurate than earlier tables.
He worked with binomial coefficients,
invented astronomical calculating machines, developed spherical trig,
and is credited with various theorems of trigonometry
including the Law of Cosines, which is sometimes called Al-Kashi's Theorem.
He is sometimes credited with the invention of decimal fractions
(though he worked mainly with sexagesimal fractions),
and a method like Horner's to calculate roots.
However decimal fractions had been used earlier, e.g. by Qin Jiushao;
and Al-Kashi's root calculations may also have been
derived from Chinese texts by Qin Jiushao or Zhu Shiejie.

Using his methods, al-Kashi calculated π correctly
to 17 significant digits, breaking Madhava's record.
(This record was subsequently broken by
relative unknowns: a German ca. 1600, John Machin 1706.
In 1949 the π calculation record was held briefly
by John von Neumann and the ENIAC.)

Regiomontanus was a prodigy who entered University at age eleven,
studied under the influential Georg von Peuerbach,
and eventually collaborated with him.
He was an important astronomer;
he found flaws in Ptolemy's system (thus influencing Copernicus),
realized lunar observations could be used to determine
longitude, and may have believed in heliocentrism.
His ephemeris was used by Columbus, when shipwrecked on Jamaica,
to predict a lunar eclipse,
thus dazzling the natives and perhaps saving his crew.
More importantly, Regiomontanus was one of the most influential mathematicians
of the Middle Ages; he published trigonometry textbooks and tables,
as well as the best textbook on arithmetic and algebra of his time.
(Regiomontanus lived shortly after Gutenberg,
and founded the first scientific press.)
He was a prodigious reader of Greek and Latin translations,
and most of his results were copied from Greek works
(or indirectly from Arabic writers, especially Jabir ibn Aflah);
however he improved or reconstructed many of the proofs, and
often presented solutions in both geometric and algebraic form.
His algebra was more symbolic and general than his predecessors';
he solved cubic equations (though not the general case);
applied Chinese remainder methods, and worked in number theory.
He posed and solved a variety of clever geometric puzzles, including
his famous angle maximization problem.
Regiomontanus was also an instrument maker, astrologer, and Catholic bishop.
He died in Rome where he had been called to
advise the Pope on the calendar; his early death may have delayed
the needed reform until the time of Pope Gregory.

Leonardo da Vinci is most renowned for his paintings --
Mona Lisa
and The Last Supper are among the most discussed
and admired paintings ever --
but he did much other work and was probably the most
talented, versatile and prolific polymath ever to live;
his writings exceed 13,000 folios.
He developed new techniques, and principles of perspective geometry,
for drawing, painting and sculpture;
he was also an expert architect and engineer;
and surely the most prolific inventor of all time.
Although most of his paper designs were never built, Leonardo's inventions
include reflecting and refracting telescope, adding machine,
parabolic compass,
improved anemometer, parachute, helicopter, flying ornithopter,
several war machines (multi-barreled gun, steam-driven cannon,
tank, giant crossbow, finned mortar shells, portable bridge),
pumps, an accurate spring-operated clock,
bobbin winder, robots, scuba gear, an elaborate musical
instrument he called the 'viola organista,' and more.
(Some of his designs, including the viola organista, his
parachute, and a large single-span bridge, were finally built five
centuries later; and worked as intended.)
He developed the mechanical theory of the arch;
made advances in anatomy, botany, and other fields of science;
developed an octant-based map projection;
and he was first to conceive of plate tectonics.
He was also a poet and musician.

He had little formal training in mathematics until he was
in his mid-40's, when he and Luca Pacioli (the other great Italian
mathematician of that era) began tutoring each other.
Despite this slow start, he did make novel achievements in mathematics:
he was first to note the simple classification of symmetry
groups on the plane, achieved interesting bisections
and mensurations, advanced the craft of descriptive geometry,
and developed an approximate solution
to the circle-squaring problem.
He was first to discover the 60-vertex shape now called "buckyball."
(Leonardo is also widely credited with the elegant two-hexagon proof
of the Pythagorean Theorem, but this authorship
appears to be a myth.)
Along with Archimedes, Alberuni, Leibniz, and J. W. von Goethe,
Leonardo da Vinci was among the greatest geniuses ever;
but none of these appears on Hart's List
of the Most Influential Persons in History:
genius doesn't imply influence.
(However, M.I.T.'s Pantheon project, using the statistics of
on-line biographies, prepared a list of the Thirty-Five (or Eighty)
Most Influential Persons in History; in addition to five (six) names already
on our list and Hart's -- Aristotle, Newton, Einstein, Galileo, Euclid (and Descartes) --
their list includes four (seven) other mathematicians
missing from Hart's list: Plato, Leonardo, Pythagoras, Archimedes
(and Thales, Pascal, Ptolemy).

Leonardo was also a writer and philosopher. Among his notable adages are
"Simplicity is the ultimate sophistication,"
and "The noblest pleasure is the joy of understanding,"
and "Human ingenuity ... will never discover any inventions more
beautiful, more simple or more practical than those of nature."

The European Renaissance developed in 15th-century
Italy, with the blossoming of great art, and
as scholars read books by great Islamic scientists like Alhazen.
The earliest of these great Italian polymaths were largely not noted
for mathematics, and Leonardo da Vinci began serious math study
only very late in life, so the best candidates for mathematical greatness
in the Italian Renaissance were foreigners.
Along with Regiomontanus from Bavaria, there was an even more famous
man from Poland.

Nicolaus Copernicus (Mikolaj Kopernik) was a polymath:
he studied law and medicine;
published poetry; contemplated astronomy;
worked professionally as a church scholar and diplomat;
and was also a painter.
He studied Islamic works on astronomy and geometry
at the University of Bologna, and eventually wrote a book of great impact.
Although his only famous theorem of mathematics (that certain trochoids are
straight lines) may have been derived from Oresme's work,
or copied from Nasir al-Tusi,
it was mathematical thought that led Copernicus to the conclusion
that the Earth rotates around the Sun.
Despite opposition from the Roman church, this discovery
led, via Galileo, Kepler and Newton, to the Scientific Revolution.
For this revolution, Copernicus is ranked #19 on Hart's List
of the Most Influential Persons in History; however I think
there are several reasons why Copernicus' importance may be exaggerated:
(1) Copernicus' system still used circles and epicycles, so it
was left to Kepler to discover the facts of elliptical orbits;
(2) he retained the notion of a sphere of fixed stars,
thus missing the unifying insight that our sun is one of many;
(3) Giordano Bruno (1548-1600), who built on Copernicus'
discovery, was a better and more influential scientist,
anticipating some of Galileo's concepts;
and (4) the Scientific Revolution didn't really get underway
until the invention of the telescope, which would have soon
led to the discovery of heliocentrism in any event.

Until the Protestant Reformation, which began about the time
of Copernicus' discovery, European scientists were reluctant to challenge
the Catholic Church and its belief in geocentrism.
Copernicus' book was published only posthumously.
It remains controversial whether earlier Islamic or Hindu mathematicians
(or even Archimedes with his The Sand Reckoner)
believed in heliocentrism, but
were also inhibited by religious orthodoxy.

Girolamo Cardano (or Jerome Cardan) was a highly
respected physician and was first to describe typhoid fever.
He was also an accomplished gambler and
chess player and wrote an early book on probability.
He was also a remarkable inventor:
the combination lock, an advanced gimbal, a
ciphering tool, and the Cardan shaft with universal joints are all
his inventions and are in use to this day.
(The U-joint is sometimes called the Cardan joint.)
He also helped develop the camera obscura.
Cardano made contributions to physics: he noted that
projectile trajectories are parabolas, and may have been first
to note the impossibility of perpetual motion machines.
He did work in philosophy, geology, hydrodynamics,
music; he wrote books on medicine and
an encyclopedia of natural science.

But Cardano is most remembered for his achievements in mathematics.
He was first to publish general solutions
to cubic and quartic equations,
and first to publish the use of complex numbers in calculations.
(Cardano's Italian colleagues deserve much credit:
Ferrari first solved the quartic, he or Tartaglia the cubic; and
Bombelli first treated the complex numbers as numbers in their own right.
Cardano may have been the last great mathematician
unwilling to deal with negative numbers: his treatment of cubic
equations had to deal with ax3 - bx + c = 0
and ax3 - bx = c as
two different cases.)
Cardano introduced binomial coefficients and the Binomial
Theorem, and introduced and solved the geometric
hypocyloid problem, as well as other geometric theorems
(e.g. the theorem underlying the 2:1 spur wheel
which converts circular to reciprocal rectilinear motion).
Cardano is credited with Cardano's Ring Puzzle,
still manufactured today and related to the
Tower of Hanoi puzzle.
(This puzzle may predate Cardano, and may even have been
known in ancient China.)
Da Vinci and Galileo may have been more influential
than Cardano, but of the three great generalists
in the century before Kepler, it seems clear that Cardano was the
most accomplished mathematician.

Cardano's life had tragic elements.
Throughout his life he was tormented that his
father (a friend of Leonardo da Vinci) married
his mother only after Cardano was born.
(And his mother tried several times to abort him.)
Cardano's reputation for gambling and aggression interfered
with his career.
He practiced astrology and was imprisoned for heresy
when he cast a horoscope for Jesus.
(This and other problems were due in part to
revenge by Tartaglia for Cardano's
revealing his secret algebra formulae.)
His son apparently murdered his own wife.
Leibniz wrote of Cardano:
"Cardano was a great man with all his faults;
without them, he would have been incomparable."

Bombelli was a talented engineer who wrote
an algebra textbook sometimes considered
one of the foremost achievements of the 16th century.
Although incorporating work by Cardano, Diophantus and
possibly Omar al-Khayyám,
the textbook was highly original and extremely influential.
Leibniz and Huygens were among many who praised his work.
Although noted for his new ideas of arithmetic, Bombelli based
much of his work on geometric ideas, and even pursued complex-number
arithmetic to an angle-trisection method.
In his textbook he introduced new symbolic notations, gave a
new square-root procedure based on continued fractions, allowed
negative and complex numbers, and gave the rules for manipulating these
new kinds of numbers.
Bombelli is often called the Inventor of Complex Numbers.

François Viète (or Franciscus Vieta) was a French
nobleman and lawyer who was a favorite of King Henry IV and eventually
became a royal privy councillor.
In one notable accomplishment he broke the Spanish diplomatic code,
allowing the French government to read Spain's messages and
publish a secret Spanish letter; this apparently led to the end
of the Huguenot Wars of Religion.

More importantly, Vieta was certainly
the best French mathematician prior to Descartes and Fermat.
He laid the groundwork for modern mathematics;
his works were the primary teaching for both Descartes and Fermat;
Isaac Newton also studied Vieta.
In his role as a young tutor Vieta used decimal numbers before they
were popularized by Simon Stevin and may have guessed
that planetary orbits were ellipses before Kepler.
Vieta did work in geometry, reconstructing and publishing proofs
for Apollonius' lost theorems, including all ten cases of the
general Problem of Apollonius.
Vieta also used his new algebraic techniques
to construct a regular heptagon.
He discovered several trigonometric identities including
a generalization of Ptolemy's Formula,
the latter (then called prosthaphaeresis)
providing a calculation shortcut similar to logarithms in that
multiplication is reduced to
addition (or exponentiation reduced to multiplication).
Vieta also used trigonometry to find real solutions
to cubic equations for which the Italian methods had
required complex-number arithmetic; he also used trigonometry
to solve a particular 45th-degree equation that had been
posed as a challenge.
Such trigonometric formulae revolutionized calculations and may
even have helped stimulate the development and use of
logarithms by Napier and Kepler.
He developed the first infinite-product formula for π.
In addition to his geometry and trigonometry, he also
found results in number theory, but
Vieta is most famous for his systematic use of decimal notation
and variable letters, for which he is sometimes called the "Father of
Modern Algebra."
(Vieta used A,E,I,O,U for unknowns and consonants for parameters;
it was Descartes who first used X,Y,Z for unknowns and A,B,C for parameters.)
In his works Vieta emphasized the relationships between algebraic
expressions and geometric constructions.
One key insight he had is that addends must be homogeneous
(i.e., "apples shouldn't be added to oranges"), a seemingly trivial idea
but which can aid intuition even today.

Descartes, who once wrote "I began where Vieta finished," is
now extremely famous, while Vieta is much less known. (He isn't
even mentioned once in Bell's famous Men of Mathematics.)
Many would now agree this is due in large measure to Descartes'
deliberate deprecations of competitors in his quest for personal glory.
(Vieta wasn't particularly humble either, calling himself
the "French Apollonius.")

Vieta's formula for π is clumsy to express
without trigonometry, even with modern notation.
Easiest may be to consider it the result of the BASIC program above.
Using this formula, Vieta constructed an approximation to π
that was best-yet by a European, though not as accurate as al-Kashi's
two centuries earlier.

Stevin was one of the greatest practical scientists
of the Late Middle Ages.
He worked with Holland's dykes and windmills;
as a military engineer he developed fortifications and systems of flooding;
he invented a carriage with sails that traveled faster than with horses
and used it to entertain his patron, the Prince of Orange.
He discovered several laws of mechanics including those
for energy conservation and hydrostatic pressure.
He lived slightly before Galileo who is now much more famous,
but Stevin discovered the equal rate of falling bodies before Galileo did;
and his explanation of tides was better than Galileo's, though still incomplete.
He was first to write on the concept of unstable equilibrium.
He invented improved accounting methods, and (though also invented
at about the same time by Chinese mathematician Zhu Zaiyu
and anticipated by Galileo's father, Vincenzo Galilei)
the equal-temperament music scale.
He also did work in descriptive geometry,
trigonometry, optics, geography, and astronomy.

In mathematics,
Stevin is best known for the notion of real numbers
(previously integers, rationals and irrationals were treated separately;
negative numbers and even zero and one were often not considered numbers).
He introduced (a clumsy form of) decimal fractions to Europe;
suggested a decimal metric
system, which was finally adopted 200 years later;
invented other basic notation like the symbol √.
Stevin proved several theorems about perspective geometry,
an important result in mechanics, and special cases of
the Intermediate Value Theorem later attributed to Bolzano and Cauchy.
Stevin's books, written in Dutch rather than Latin,
were widely read and hugely influential.
He was a very key figure in the development of modern European mathematics,
and may belong on our List.

Napier was a Scottish Laird who was a noted theologian
and thought by many to be a magician
(his nickname was Marvellous Merchiston).
Today, however, he is best known
for his work with logarithms, a word he invented.
(Several others, including Archimedes, had anticipated the use of logarithms.)
He published the first large table
of logarithms and also helped popularize usage of the decimal point
and lattice multiplication.
He invented Napier's Bones,
a crude hand calculator which could be used for division and
root extraction, as well as multiplication.
He also had inventions outside mathematics, especially
several different kinds of war machine.

Napier's noted textbooks also contain an exposition
of spherical trigonometry.
Although he was certainly very clever (and had novel
mathematical insights not mentioned in this summary),
Napier proved no deep theorem
and may not belong in the Top 100.
Nevertheless, his revolutionary methods of arithmetic had
immense historical importance; his logarithm tables were used
by Johannes Kepler himself, and led to the Scientific Revolution.

Galileo discovered the laws of inertia (including rudimentary
forms of all three of Newton's laws of motion), falling bodies (including
parabolic trajectories), and the pendulum; he also introduced the notion
of relativity which later physicists found so fruitful.
(Although he admired Galileo greatly, Einstein's famous result
was somewhat misnamed: it depended on the absoluteness,
not relativity, of the speed of light.)
Galileo discovered important principles of dynamics, including the
essential notion that the vector sum of forces produce an
acceleration.
(Aristotle seems not to have considered the notion of acceleration,
though his successor Strato of Lampsacus did write on it.)
Galileo may have been first to note that a larger body has less
relative cohesive strength than a smaller body.
He was a great inventor: in addition to being first to conceive
of the pendulum clock, he developed a new type of pump,
the first compound-lens microscope,
and the best telescope, thermometer, hydrostatic balance,
and cannon sector of his day.
As a famous astronomer, Galileo
pointed out that Jupiter's Moons, which he discovered, provide
a natural clock and allow a universal time to be determined
by telescope anywhere on Earth.
(This was of little use in ocean navigation since a ship's rocking
prevents the required delicate observations.
Galileo tried to measure the speed of light, but it was too fast for
him. However 66 years after Galileo discovered Jupiter's moons
and proposed using them as a clock,
the astronomer Roemer inferred the speed of light from that 'clock':
the clock had a discrepancy of up to seven minutes depending
on the Earth-Jupiter distance.)
Galileo's other astronomical discoveries also included sunspots
and lunar craters.
His discovery that Venus, like the Moon, had phases was the
critical fact which forced acceptance of Copernican heliocentrism.
Galileo's contributions outside physics and astronomy
were also enormous:
He made discoveries with the microscope he invented, and
made several important contributions to the early
development of biology.
Perhaps Galileo's most important contribution
was the Doctrine of Uniformity, the postulate that
there are universal laws of mechanics, in contrast to
Aristotelian and religious notions
of separate laws for heaven and earth.

Galileo is often called the "Father of Modern Science"
because of his emphasis on experimentation.
His use of a ramp to discover his Law of Falling Bodies
was ingenious.
(For his experiments he started with a water-clock to measure time,
but found the beats reproduced by trained musicians to be more convenient.)
He understood that results needed to be repeated and averaged
(he minimized mean absolute-error for his curve-fitting
criterion, two centuries before Gauss and Legendre introduced
the mean squared-error criterion).
For his experimental methods and discoveries,
his laws of motion, and for (eventually) helping
to spread Copernicus' heliocentrism, Galileo may have been
the most influential scientist ever; he
ranks #12 on Hart's list of the Most Influential Persons in History.
(Despite these comments, it does appear that Galileo ignored
experimental results that conflicted with his theories.
For example, the Law of the Pendulum, based on Galileo's incorrect
belief that the tautochrone was the circle, conflicted
with his own observations.
Some of his other ideas were wrong; for example, he dismissed Kepler's
elliptical orbits and notion of gravitation and published a
very faulty explanation of tides.)
Despite his extreme importance to mathematical physics,
Galileo doesn't usually appear on
lists of greatest mathematicians.
However, Galileo did do work in pure mathematics;
he derived certain centroids and the parabolic shape
of trajectories using a rudimentary calculus,
and mentored Bonaventura Cavalieri, who extended Galileo's calculus;
he named (and may have been first to discover) the cycloid curve.
Moreover, Galileo was one of the first to write about
infinite equinumerosity (the "Hilbert's Hotel Paradox").
Galileo once wrote "Mathematics is the language in which God has written the
universe."

Kepler was interested in astronomy from an early age,
studied to become a Lutheran minister,
became a professor of mathematics instead, then Tycho Brahe's understudy,
and, on Brahe's death, was appointed Imperial Mathematician at
the age of twenty-nine.
His observations of the planets with Brahe, along with his study
of Apollonius' 1800-year old work, led to Kepler's three Laws
of Planetary Motion, which in turn led directly to Newton's
Laws of Motion.
Beyond his discovery of these Laws (one of the most important
achievements in all of science), Kepler is also sometimes called
the "Founder of Modern Optics."
He furthered the theory of the camera obscura,
telescopes built from two convex lenses,
and atmospheric refraction.
The question of human vision had been
considered by many great scientists including Aristotle, Euclid,
Ptolemy, Galen, Alkindus, Alhazen, and Leonardo da Vinci,
but it was Kepler who was first to explain the
operation of the human eye correctly and to note that retinal
images will be upside-down.
Kepler developed a rudimentary notion of universal gravitation,
and used it to produce the best explanation for tides before Newton;
however he seems not to have noticed that his empirical laws
implied inverse-square gravitation.
Kepler ranks #75 on Michael Hart's famous list of
the Most Influential Persons in History. This rank, much lower than
that of Copernicus, Galileo or Newton, seems to me to underestimate Kepler's
importance, since it was Kepler's Laws, rather than just heliocentrism,
which were essential to the early development of mathematical physics.

According to Kepler's Laws, the planets move at variable speed
along ellipses.
(Even Copernicus thought the orbits could be described with only circles.)
The Earth-bound observer is himself describing
such an orbit and in almost the same plane as the planets;
thus discovering the Laws would be a difficult challenge even for
someone armed with computers and modern mathematics.
(The very famous Kepler Equation relating a planet's
eccentric and anomaly is just one tool Kepler needed to develop.)
Kepler understood the importance of his remarkable discovery, even if
contemporaries like Galileo did not, writing:

"I give myself up to divine ecstasy ... My book is written.
It will be read either by my contemporaries or by posterity —
I care not which. It may well wait a hundred years for a
reader, as God has waited 6,000 years for someone to
understand His work."

Kepler also once wrote "Mathematics is the archetype of the
beautiful."

Besides the trigonometric results needed to discover his Laws,
Kepler made other contributions to mathematics.
He generalized Alhazen's Billiard Problem, developing the
notion of curvature.
He was first to notice that the set of Platonic regular solids
was incomplete if concave solids are admitted, and first
to prove that there were only 13 Archimedean solids.
He proved theorems of solid geometry later discovered on
the famous palimpsest of Archimedes.
He rediscovered the Fibonacci series, applied it to
botany, and noted that the ratio of Fibonacci numbers
converges to the Golden Mean.
He was a key early pioneer in calculus,
and embraced the concept of continuity
(which others avoided due to Zeno's paradoxes);
his work was a direct inspiration for Cavalieri and others.
He developed the theory of logarithms and improved on
Napier's tables.
He developed mensuration methods and
anticipated Fermat's theorem on stationary points.
Kepler once had an opportunity to buy wine, which merchants measured
using a shortcut; with the famous Kepler's Wine Barrel Problem,
he used his rudimentary calculus to deduce which barrel shape
would be the best bargain.

Kepler reasoned that the structure of snowflakes was evidence
for the then-novel atomic theory of matter.
He noted that the obvious packing of cannonballs gave maximum
density (this became known as Kepler's Conjecture; optimality
was proved among regular packings by Gauss, but it wasn't until
1998 that the possibility of denser irregular packings
was disproven).
In addition to his physics and mathematics,
Kepler wrote a science fiction novel,
and was an astrologer and mystic.
He had ideas similar to Pythagoras about numbers ruling the
cosmos (writing that the purpose of studying the world
"should be to discover the rational order and harmony which
has been imposed on it by God and which He revealed to us in the language of
mathematics").
Kepler's mystic beliefs even led to his own mother being
imprisoned for witchcraft.

Johannes Kepler (along with Galileo, Fermat,
Huygens, Wallis, Vieta and Descartes)
is among the giants on whose shoulders Newton was proud to stand.
Some historians place him ahead of Galileo and Copernicus as the single most
important contributor to the early Scientific Revolution.
Chasles includes Kepler on a list of the six responsible for
conceiving and perfecting infinitesimal calculus (the other five are
Archimedes, Cavalieri, Fermat, Leibniz and Newton).
(www.keplersdiscovery.com
is a wonderful website devoted to Johannes Kepler's discoveries.)

Desargues invented projective geometry and found the relationship
among conic sections which inspired Blaise Pascal.
Among several ingenious and rigorously proven theorems
are Desargues' Involution Theorem
and his Theorem of Homologous Triangles.
Desargues was also a noted architect and inventor:
he produced an elaborate spiral staircase, invented an
ingenious new pump based on the epicycloid, and had
the idea to use cycloid-shaped teeth in the design of gears.

Desargues' projective geometry may have been too creative for his
time; Descartes admired Desargues but was disappointed his friend
didn't apply algebra to his geometric results as Descartes did;
Desargues' writing was poor; and one of his best pupils
(Blaise Pascal himself) turned away from math, so Desargues' work
was largely ignored (except by Philippe de La Hire, Desargues'
other prize pupil) until Poncelet rediscovered it almost two centuries later.
(Copies of Desargues' own works surfaced about the same time.)
For this reason, Desargues may not be important enough
to belong in the Top 100, despite that he may have been among the
greatest natural geometers ever.

Descartes' early career was that of soldier-adventurer
and he finished as tutor to royalty, but in between
he achieved fame as the preeminent intellectual of his day.
He is considered the inventor of both analytic geometry and
symbolic algebraic notation and is therefore
called the "Father of Modern Mathematics."
His use of equations to partially solve the geometric Problem of Pappus
revolutionized mathematics.
Because of his famous philosophical writings ("Cogito ergo sum") he
is considered, along with Aristotle, to be one of the
most influential thinkers in history.
He ranks #49 on Michael Hart's famous list of
the Most Influential Persons in History.
His famous mathematical theorems include the
Rule of Signs (for determining the signs of
polynomial roots), the elegant formula
relating the radii of Soddy kissing circles,
his theorem on total angular defect (an early form of
the Gauss-Bonnet result so key to much mathematics),
and an improved solution to the Delian problem (cube-doubling).
While studying lens refraction,
he invented the Ovals of Descartes.
He improved mathematical notation
(e.g. the use of superscripts to denote exponents).
He also discovered Euler's Polyhedral Theorem, F+V = E+2.
Descartes was very influential in physics and biology as well,
e.g. developing laws of motion which included a "vortex" theory of gravitation;
but most of his scientific work outside mathematics was eventually
found to be incorrect.

Descartes has an extremely high reputation and would
be ranked even higher by many list makers, but whatever his
historical importance his mathematical skill was not in the
top rank. Some of his work was borrowed from others,
e.g. from Thomas Harriot.
He had only insulting things to say about Pascal and Fermat,
each of whom was much more
brilliant at mathematics than Descartes.
(Some even suspect that Descartes arranged the destruction
of Pascal's lost Essay on Conics.)
And Descartes made numerous errors in his development of
physics, perhaps even delaying science,
with Huygens writing "in all of [Descartes'] physics,
I find almost nothing to which I can subscribe as being correct."
Even the historical importance of his mathematics
may be somewhat exaggerated since others, e.g.
Fermat, Wallis and Cavalieri, were making similar discoveries
independently.

Cavalieri worked in analysis, geometry
and trigonometry (e.g. discovering a formula for the area of a spherical
triangle), but is most famous for publishing
works on his "principle of indivisibles" (calculus);
these were very influential
and inspired further development by Huygens, Wallis and Barrow.
(His calculus was partly anticipated by Galileo, Kepler and Luca Valerio,
and developed independently, though left unpublished, by Fermat.)
Among his theorems in this calculus was
lim(n→∞)
(1m+2m+ ... +nm)
/ nm+1 = 1 / (m+1)Cavalieri also worked in theology, astronomy, mechanics and optics;
he was an inventor, and published logarithm tables.
He wrote several books, the first one developing the
properties of mirrors shaped as conic sections.
His name is especially remembered for Cavalieri's
Principle of Solid Geometry.
Galileo said of Cavalieri, "Few, if any, since Archimedes, have delved
as far and as deep into the science of geometry."

Pierre de Fermat was the most brilliant mathematician
of his era and, along with Descartes, one of the most influential.
Although mathematics was just his hobby (Fermat was a government lawyer),
Fermat practically founded Number Theory, and also played key
roles in the discoveries of Analytic Geometry and Calculus.
Lagrange considered Fermat, rather than Newton or Leibniz, to be the
inventor of calculus.
Fermat was first to study certain interesting curves,
e.g. the "Witch of Agnesi".
He was also an excellent geometer (e.g. discovering a
triangle's Fermat point),
and (in collaboration with Blaise Pascal) discovered probability theory.
Fellow geniuses are the best judges of genius, and Blaise
Pascal had this to say of Fermat:
"For my part, I confess that [Fermat's researches about numbers]
are far beyond me, and I am competent only to admire them."
E.T. Bell wrote "it can be argued that Fermat was
at least Newton's equal as a pure mathematician."

Fermat's most famous discoveries in number theory
include the ubiquitously-used Fermat's Little Theorem;
the n = 4 case of
his conjectured Fermat's Last Theorem (he may have
proved the n = 3 case as well);
and Fermat's Christmas Theorem
(that any prime (4n+1) can be represented as the sum of two
squares in exactly one way) which may be considered the
most difficult theorem of arithmetic which had been
proved up to that date.
Fermat proved the Christmas Theorem with difficulty
using "infinite descent," but details are unrecorded, so the
theorem is often named the Fermat-Euler Prime Number Theorem,
with the first published proof being by Euler more than a century after
Fermat's claim.
Another famous conjecture by Fermat is that
every natural number is the sum of three triangle numbers,
or more generally the sum of k k-gonal numbers.
As with his "Last Theorem" he claimed to have a proof but
didn't write it up.
(This theorem was eventually proved by Lagrange for k=4,
the very young Gauss for k=3, and Cauchy for general k.
Diophantus claimed the k=4 case but any proof has been lost.)
I think Fermat's conjectures were impressive even if unproven,
and that this great mathematician is often underrated.
(Recall that his so-called "Last Theorem"
was actually just a private scribble.)

Fermat developed a system of analytic geometry which both
preceded and surpassed that of Descartes; he developed methods of
differential and integral calculus which Newton
acknowledged as an inspiration.
Although Kepler anticipated it, Fermat is credited with Fermat's
Theorem on Stationary Points (df(x)/dx = 0 at function extrema),
the key to many problems in applied analysis.
Fermat was also the first European to find the integration
formula for the general polynomial; he used his calculus
to find centers of gravity, etc.

Fermat's contemporaneous rival René Descartes is more famous
than Fermat, and Descartes' writings were more influential.
Whatever one thinks of Descartes as a philosopher,
however, it seems clear that Fermat was the better mathematician.
Fermat and Descartes did work in physics and independently discovered
the (trigonometric) law of refraction,
but Fermat gave the correct explanation, and used it
remarkably to anticipate the Principle of Least Action
later enunciated by Maupertuis (though Maupertuis
himself, like Descartes, had an incorrect explanation of refraction).
Fermat and Descartes independently discovered analytic geometry,
but it was Fermat who extended it to more than
two dimensions, and followed up by developing elementary calculus.

Roberval was an eccentric genius, underappreciated because
most of his work was published only long after his death.
He did early work in integration, following Archimedes rather
than Cavalieri; he worked on analytic geometry independently of Descartes.
With his analysis he was able to solve several difficult geometric problems
involving curved lines and solids, including results about the
cycloid which were also credited to Pascal and Torricelli.
Some of these methods, published posthumously, led to him being called the
Founder of Kinematic Geometry.
He excelled at mechanics,
worked in cartography, helped Pascal with vacuum experiments, and invented
the Roberval balance, still in use in weighing scales to this day.
He opposed Huygens in the early debate about gravitation,
though neither fully anticipated Newton's solution.

Torricelli was a disciple of Galileo (and succeeded him
as grand-ducal mathematician of Tuscany).
He was first to understand that a barometer measures atmospheric
weight, and used this insight to invent the mercury barometer and
to create a sustained vacuum (then thought impossible).
He was a skilled craftsman who built
the best telescopes and microscopes of his day.
As mathematical physicist, he extended Galileo's results, was
first to explain winds correctly, and discovered several key principles
including Torricelli's Law (water drains through a small hole with rate
proportional to the square root of water depth).
In mathematics, he applied Cavalieri's methods to
solve difficult mensuration problems; he also wrote on possible pitfalls
in applying the new calculus.
He discovered Gabriel's Horn with infinite surface area but finite
volume; this "paradoxical" result provoked much discussion at the time.
He also solved a problem due to Fermat by locating the isogonic center of
a triangle.
Torricelli was a significant influence on the early scientific revolution;
had he lived longer, or published more, he would surely have become
one of the greatest mathematicians of his era.

Wallis began his life as a savant at arithmetic
(it is said he once calculated the square root of a 53-digit
number to help him sleep and remembered the result in the morning),
a medical student (he may have contributed to the concept
of blood circulation), and theologian, but went on to become perhaps the most
brilliant and influential English mathematician before Newton.
He made major advances in analytic geometry, but also
contributions to algebra, geometry and trigonometry.
Unlike his contemporary Huygens, who took inspiration from
Euclid's rigorous geometry, Wallis embraced the new analytic methods
of Descartes and Fermat.
He is especially famous for using negative and fractional exponents
(though Oresme had introduced fractional exponents three centuries earlier),
taking the areas of curves, and treating inelastic collisions
(he and Huygens were first to develop the law of momentum conservation).
He was the first European to solve Pell's Equation.
Like Vieta, Wallis was a code-breaker, helping the Commonwealth
side (though he later petitioned against the beheading of King Charles I).
He was the first great mathematician to consider complex numbers legitimate;
and first to use the symbol ∞.
Wallis coined several terms including continued fraction,
induction, interpolation, mantissa, and hypergeometric series.

Also like Vieta, Wallis created an infinite product formula for pi,
which might be (but isn't) written today as:
π = 2 ∏k=1,∞ 1+(4k2-1)-1

Pascal was an outstanding genius who studied geometry as a child.
At the age of sixteen he stated and proved Pascal's Theorem, a
fact relating any six points on any conic section.
The Theorem is sometimes called the "Cat's Cradle"
or the "Mystic Hexagram."
Pascal followed up this result by showing that each of Apollonius'
famous theorems about conic sections was a corollary of the Mystic Hexagram;
along with Gérard Desargues (1591-1661),
he was a key pioneer of projective geometry.
He also made important early contributions to calculus;
indeed it was his writings that inspired Leibniz.
Returning to geometry late in life, Pascal advanced the theory of the cycloid.
In addition to his work in geometry and calculus, he
founded probability theory, and made contributions
to axiomatic theory.
His name is associated with the Pascal's Triangle
of combinatorics and Pascal's Wager in theology.

Like most of the greatest mathematicians, Pascal was interested in
physics and mechanics, studying fluids, explaining vacuum,
and inventing the syringe and hydraulic press.
At the age of eighteen he designed and built the world's first automatic
adding machine.
(Although he continued to refine this invention,
it was never a commercial success.)
He suffered poor health throughout his life,
abandoned mathematics for religion at about age 23,
wrote the philosophical treatise Pensées
("We arrive at truth, not by reason only, but also by the heart"),
and died at an early age.
Many think that had he devoted more years to mathematics,
Pascal would have been one of the greatest mathematicians ever.

Christiaan Huygens (or Hugens, Huyghens) was second
only to Newton as the greatest mechanist of his era.
Although an excellent mathematician, he
is much more famous for his physical theories and inventions.
He developed laws of motion before Newton, including
the inverse-square law of gravitation, centripetal force,
and treatment of solid bodies rather than point approximations;
he (and Wallis) were first to state the law of momentum conservation correctly.
He advanced the wave ("undulatory") theory of light,
a key concept being Huygen's Principle, that each point on a wave front
acts as a new source of radiation.
His optical discoveries include explanations for polarization
and phenomena like haloes.
(Because of Newton's high reputation and
corpuscular theory of light, Huygens' superior wave theory
was largely ignored until the 19th-century work of Young, Fresnel,
and Maxwell.
Later, Planck, Einstein and Bohr, partly anticipated by Hamilton,
developed the modern notion of wave-particle duality.)

Huygens is famous for his inventions of clocks and lenses.
He invented the escapement and other mechanisms, leading to the
first reliable pendulum clock; he built the first
balance spring watch, which he presented to his
patron, King Louis XIV of France; he was first to give the
correct "equation of time" relating sundial time to
absolute time.
He invented superior lens grinding techniques,
the achromatic eye-piece, and the best telescope of his day.
He was himself a famous astronomer:
he discovered Titan and was first to
properly describe Saturn's rings and the Orion Nebula.
He also designed, but never built, an internal combustion engine.
He promoted the use of an equal-tempered 31-tone music scale
to avoid the tuning errors in Stevin's 12-tone scale;
a 31-tone organ was in use in Holland as late as the 20th century.
Huygens was an excellent card player, billiard player,
horse rider, and wrote a book speculating about extra-terrestrial life.

As a mathematician, Huygens did brilliant work in analysis;
his calculus, along with that of Wallis, is considered the
best prior to Newton and Leibniz.
He also did brilliant work in geometry,
proving theorems about conic sections, the cycloid and the catenary.
He was first to show that the cycloid solves the tautochrone
problem; he used this fact to design
pendulum clocks that would be more accurate
than ordinary pendulum clocks.
He was first to find the flaw in Saint-Vincent's
then-famous circle-squaring method;
Huygens himself solved some related quadrature problems.
He introduced the concepts of evolute and involute.
His friendships with Descartes, Pascal, Mersenne and others helped
inspire his mathematics; Huygens in turn was inspirational
to the next generation.
At Pascal's urging, Huygens published the first real
textbook on probability theory; he also
became the first practicing actuary.

Huygens had tremendous creativity, historical importance,
and depth and breadth of genius, both in physics and mathematics.
He also was important for serving as tutor
to the otherwise self-taught Gottfried Leibniz
(who'd "wasted his youth" without learning any math).
Before agreeing to tutor him,
Huygens tested the 25-year old Leibniz by asking him
to sum the reciprocals of the triangle numbers.

Seki Takakazu (aka Shinsuke) was a self-taught prodigy who
developed a new notation for algebra, and made several discoveries
before Western mathematicians did; these
include determinants, the Newton-Raphson method,
Newton's interpolation formula,
Bernoulli numbers, discriminants, methods of calculus,
and probably much that has been forgotten
(Japanese schools practiced secrecy).
He calculated π to ten decimal places using
Aitkin's method (rediscovered in the 20th century).
He also worked with magic squares.
He is remembered as a brilliant genius and very influential teacher.

Seki's work was not propagated to Europe, so has minimal
historic importance; otherwise Seki might rank high on our list.

James Gregory (Gregorie) was the outstanding Scottish genius of his century.
Had he not died at the age of 36, or if he had published more of his work,
(or if Newton had never lived,) Gregory would surely be appreciated
as one of the greatest mathematicians of the early Age of Science.
Inspired by Kepler's work, he worked in mechanics and optics; invented
a reflecting telescope; and is even
credited with using a bird feather as the first diffraction grating.
But James Gregory is most famous for his mathematics, making
many of the same discoveries as Newton did:
the Fundamental Theorem of Calculus,
interpolation method, and binomial theorem.
He developed the concept of Taylor's series and used it to
solve a famous semicircle division problem posed by Kepler
and to develop trigonometric identities, including
tan-1x = x - x3/3 + x5/5 - x7/7 + ... (for |x| < 1)
Gregory anticipated Cauchy's convergence test, Newton's identities
for the powers of roots, and Riemann integration.
He may have been first to suspect that quintics generally lacked
algebraic solutions, as well as that π and e were
transcendental.
He produced a partial proof that the ancient "Squaring the Circle"
problem was impossible.

Gregory declined to publish much of his work, partly in deference
to Isaac Newton who was making many of the same discoveries.
Because the wide range of his mathematics wasn't appreciated until long after
his death, Gregory lacks the historic importance to qualify for the Top 100.

Newton was an industrious lad who built marvelous toys
(e.g. a model windmill powered by a mouse on treadmill).
At about age 22, on leave from University, this genius began
revolutionary advances in mathematics, optics, dynamics,
thermodynamics, acoustics and celestial mechanics.
He is famous for his Three Laws of Motion
(inertia, force, reciprocal action) but, as
Newton himself acknowledged, these Laws weren't fully novel:
Hipparchus, Ibn al-Haytham, Descartes, Galileo and Huygens had all
developed much basic mechanics already; and Newton credits the First Law
to Aristotle.
However Newton was apparently the first person to conclude
that the ordinary gravity we observe on Earth is the very same
force that keeps the planets in orbit.
His Law of Universal Gravitation was revolutionary and due to
Newton alone.
(Christiaan Huygens, the other great mechanist of the era,
had independently deduced that Kepler's laws imply inverse-square
gravitation, but he considered the action at a distance in
Newton's theory to be "absurd.")
Newton published the Cooling Law of thermodynamics.
He also made contributions to chemistry, and was
the important early advocate of the atomic theory.
His writings also made important contributions
to the general scientific method.
His other intellectual interests included theology,
and mysticism.
He studied ancient Greek writers like Pythagoras,
Democritus, Lucretius, Plato; and claimed that the
ancients knew much, including the law of gravitation.

Although this list is concerned only with mathematics,
Newton's greatness is indicated by the huge range of his
physics: even without his Laws of Motion, Gravitation and Cooling,
he'd be famous just for his revolutionary work in optics, where
he explained diffraction, observed
that white light is a mixture of all the rainbow's colors,
noted that purple is created by combining red and
blue light and, starting from that observation,
was first to conceive of a color hue "wheel."
(The mystery of the rainbow had been solved by earlier
mathematicians like Al-Farisi and Descartes, but
Newton improved on their explanations. Most people
would count only six colors in the rainbow but, due to
Newton's influence, seven -- a number with mystic importance --
is the accepted number. Supernumerary rainbows, by the way,
were not explained until the wave theory of light superceded
Newton's theory.)
He noted that his dynamical laws were symmetric in time; that
just as the past determines the future, so the
future might, in principle, determine the past.
Newton almost anticipated Einstein's mass-energy equivalence,
writing "Gross Bodies and Light are convertible into one another...
[Nature] seems delighted with Transmutations."
Ocean tides had intrigued several of Newton's predecessors; once
gravitation was known, the Moon's gravitational attraction
provided the explanation -- except that there are two high
tides per day, one when the Moon is farthest away.
With clear thinking the second high tide is also explained by gravity
but who was the first clear thinker to produce that explanation?
You guessed it! Isaac Newton.
(The theory of tides was later refined by Laplace.)
Newton's earliest fame came when he designed the first reflecting telescope:
by avoiding chromatic aberration, these were the best telescopes of that era.
He also designed the first reflecting microscope, and the sextant.

Although others also developed the techniques independently,
Newton is regarded as the "Father of Calculus" (which he called
"fluxions"); he shares credit with Leibniz for the
Fundamental Theorem of Calculus
(that integration and differentiation are each other's inverse operation).
He applied calculus for several purposes:
finding areas, tangents, the lengths of
curves and the maxima and minima of functions.
Although Descartes is renowned as the inventor of analytic geometry,
he and followers like Wallis were reluctant even to use negative
coordinates, so one historian declares Newton to be "the first
to work boldly with algebraic equations."
In addition to several other important advances in analytic geometry,
his mathematical works include the Binomial Theorem,
his eponymous interpolation method,
the idea of polar coordinates,
and power series for exponential and trigonometric functions.
(His equation
ex = ∑ xk / k!
has been called the "most important series in mathematics.")
He contributed to algebra and the theory of equations;
he was first to state Bézout's Theorem;
he generalized Descartes' rule of signs.
(The generalized rule of signs was incomplete and finally
resolved two centuries later by Sturm and Sylvester.)
He developed a series for the arcsin function.
He developed facts about cubic equations
(just as the "shadows of a cone" yield all quadratic curves,
Newton found a curve whose "shadows" yield all cubic curves).
He proved, using a purely geometric argument of awesome ingenuity,
that same-mass spheres (or hollowed spheres) of any radius have equal
gravitational attraction: this fact is key to celestial motions.
(He also proved that objects inside a hollowed sphere
experience zero net attraction.)
He discovered Puiseux series almost two centuries before they
were re-invented by Puiseux.
(Like some of the greatest ancient mathematicians,
Newton took the time to compute an approximation to
π; his was better than Vieta's, though still
not as accurate as al-Kashi's.)

Newton is so famous for his calculus, optics, and laws of
gravitation and motion, it is easy to overlook that he was also one of the
very greatest geometers.
He was first to fully solve
the famous Problem of Pappus, and did so with pure geometry.
Building on the "neusis" (non-Platonic) constructions of Archimedes
and Pappus, he demonstrated cube-doubling and that angles
could be k-sected for any k, if one is allowed a conchoid or
certain other mechanical curves.
He also built on Apollonius' famous
theorem about tangent circles to develop the technique
now called hyperbolic trilateration.
Despite the power of Descartes' analytic geometry,
Newton's achievements with synthetic geometry were surpassing.
Even before the invention of the calculus of variations, Newton
was doing difficult work in that field, e.g.
his calculation of the "optimal bullet shape."
His other marvelous geometric theorems included several about quadrilaterals
and their in- or circum-scribing ellipses.
He constructed the parabola defined by four given points,
as well as various cubic curve constructions.
(As with Archimedes, many of
Newton's constructions used non-Platonic tools.)
He anticipated Poncelet's Principle of Continuity.
An anecdote often cited to demonstrate his brilliance is the problem
of the brachistochrone, which had baffled the best mathematicians in
Europe, and came to Newton's attention late in life.
He solved it in a few hours and published the answer anonymously.
But on seeing the solution Jacob Bernoulli immediately exclaimed
"I recognize the lion by his footprint."

In 1687 Newton published
Philosophiae Naturalis Principia Mathematica, surely
the greatest scientific book ever written.
The motion of the planets was not understood before Newton,
although the heliocentric system allowed Kepler to describe the
orbits.
In Principia Newton analyzed the consequences of his Laws of Motion
and introduced the Law of Universal Gravitation.
With the key mystery of celestial motions finally resolved,
the Great Scientific Revolution began.
(In his work Newton also proved important theorems about inverse-cube
forces, work largely unappreciated until Chandrasekhar's modern-day work.)
Newton once wrote "Truth is ever to be found in the simplicity,
and not in the multiplicity and confusion of things."
Sir Isaac Newton was buried at Westminster Abbey in a tomb
inscribed "Let mortals rejoice that so great an ornament to the human
race has existed."

Newton ranks #2 on Michael Hart's famous list of
the Most Influential Persons in History.
(Muhammed the Prophet of Allah is #1.)
Whatever the criteria, Newton would certainly rank first
or second on any list of physicists, or scientists in general,
but some listmakers would demote him slightly on a list of
pure mathematicians:
his emphasis was physics not mathematics,
and the contribution of Leibniz
(Newton's rival for the title Inventor of Calculus)
lessens the historical importance of Newton's calculus.
One reason I've ranked him at #1 is a comment by
Gottfried Leibniz himself:
"Taking mathematics from the beginning of the world to the time when
Newton lived, what he has done is much the better part."

Leibniz was one of the most brilliant and prolific
intellectuals ever; and his influence in mathematics (especially
his co-invention of the infinitesimal calculus) was immense.
His childhood IQ has been estimated as second-highest in all of history,
behind only Goethe's.
Descriptions which have been applied to Leibniz include
"one of the two greatest universal geniuses" (da Vinci was
the other); "the most important logician between Aristotle and Boole;"
and the "Father of Applied Science."
Leibniz described himself as "the most teachable of mortals."

Mathematics was just a self-taught sideline for Leibniz, who was
a philosopher, lawyer, historian, diplomat and renowned inventor.
Because he "wasted his youth" before learning mathematics,
he probably ranked behind the Bernoullis as well as Newton
in pure mathematical talent, and thus he may be the only
mathematician among the Top Fifteen who was never the greatest
living algorist or theorem prover.
I won't try to summarize Leibniz' contributions to philosophy
and diverse other fields including biology; as just
three examples: he predicted the Earth's molten core,
introduced the notion of subconscious mind,
and built the first calculator that could do multiplication.
Leibniz also had political influence: he
consulted to both the Holy Roman and Russian Emperors;
another of his patrons was Sophia Wittelsbach (Electress of Hanover),
who was only distantly in
line for the British throne, but was made Heir Presumptive.
(Sophia died before Queen Anne, but her son
was crowned King George I of England.)

Leibniz pioneered the common discourse of mathematics,
including its continuous, discrete, and symbolic aspects.
(His ideas on symbolic logic weren't pursued and it was left
to Boole to reinvent this almost two centuries later.)
Mathematical innovations attributed to Leibniz include
the notations ∫f(x)dx,
df(x)/dx, ∛x,
and even the use of a·b (instead of
a X b) for multiplication;
the concepts of matrix determinant and Gaussian elimination;
the theory of geometric envelopes;
and the binary number system.
He worked in number theory, conjecturing Wilson's Theorem.
He invented more mathematical terms than anyone, including
function, analysis situ, variable, abscissa,
parameter and coordinate.
He also coined the word transcendental, proving that
sin() was not an algebraic function.
His works seem to anticipate cybernetics and information theory;
and Mandelbrot acknowledged Leibniz' anticipation of self-similarity.
Like Newton, Leibniz discovered The Fundamental Theorem of Calculus;
his contribution to calculus was much more influential than Newton's,
and his superior notation is used to this day.
As Leibniz himself pointed out, since the concept of
mathematical analysis was already known to ancient Greeks,
the revolutionary invention was the notation ("calculus"),
because with "symbols [which] express the exact nature of a
thing briefly ... the labor of thought is wonderfully diminished."

Leibniz' thoughts on mathematical physics had some influence.
He was one of the first to articulate the law of energy
conservation and may have written on the principle of least action.
He developed laws of motion that gave different insights
from those of Newton; his views on cosmology anticipated theories
of Mach and Einstein and are more in accord with modern
physics than are Newton's views.
Mathematical physicists influenced by Leibniz include not only Mach,
but perhaps Hamilton and Poincaré themselves.

Jacob Bernoulli studied the works of Wallis and Barrow;
he and Leibniz became friends and tutored each other.
Jacob developed important methods for integral
and differential equations, coining the word integral.
He and his brother were the key pioneers in mathematics during the generations
between the era of Newton-Leibniz and the rise of Leonhard Euler.

Jacob liked to pose and solve physical optimization problems.
His "catenary" problem (what shape does a clothesline take?)
became more famous than the "tautochrone" solved by Huygens.
Perhaps the most famous of such problems
was the brachistochrone, wherein Jacob recognized
Newton's "lion's paw", and about which Johann Bernoulli wrote:
"You will be petrified with astonishment [that]
this same cycloid, the tautochrone of Huygens,
is the brachistochrone we are seeking."
Jacob did significant work outside calculus;
in fact his most famous work was the Art of Conjecture,
a textbook on probability and combinatorics which
proves the Law of Large Numbers, the Power Series Equation,
and introduces the Bernoulli numbers.
He is credited with the invention of polar coordinates (though
Newton and Alberuni had also discovered them).
Jacob also did outstanding work in geometry, for example constructing
perpendicular lines which quadrisect a triangle.

Johann Bernoulli learned from his older brother and Leibniz,
and went on to become principal teacher to Leonhard Euler.
He developed exponential calculus;
together with his brother Jacob, he founded the
calculus of variations.
Johann solved the catenary before Jacob did;
this led to a famous rivalry in the Bernoulli family.
(No joint papers were written; instead the Bernoullis,
especially Johann, began claiming each others' work.)
Although his older brother may have demonstrated greater breadth,
Johann had no less skill than Jacob,
contributed more to calculus,
discovered L'Hôpital's Rule before L'Hôpital did,
and made important contributions in physics, e.g. about
vibrations, elastic bodies, optics, tides, and ship sails.

It may not be clear which Bernoulli was the "greatest."
Johann has special importance as tutor to Leonhard Euler,
but Jacob has special importance as tutor to his brother Johann.
Johann's son Daniel is also a candidate for greatest Bernoulli.

De Moivre was an important pioneer of analytic geometry
and, especially, probability theory.
(He and Laplace may be regarded as the two most important early
developers of probability theory.)
In probability theory he developed actuarial science, posed interesting
problems (e.g. about derangements), discovered the normal
and Poisson distributions, and proposed (but didn't prove) the
Central Limit Theorem.
He was first to discover a closed-form formula for the Fibonacci numbers;
and he developed an early version of Stirling's approximation to n!.
He discovered De Moivre's Theorem:
(cos x + i sin x)n
= cos nx + i sin nx

He was a close friend and muse of Isaac Newton, who allegedly
told people who asked about Principia:
"Go to Mr. De Moivre; he knows these things better than I do."

Brook Taylor invented integration by parts, developed what is now
called the calculus of finite differences, developed
a new method to compute logarithms, made several other key discoveries
of analysis, and did significant work in mathematical physics.
His love of music and painting may have motivated some of his
mathematics:
He studied vibrating strings; and
also wrote an important treatise on perspective in drawing
which helped develop the fields of both projective
and descriptive geometry.
His work in projective geometry rediscovered Desargues' Theorem, introduced
terms like vanishing point, and influenced Lambert.

Taylor was one of the few mathematicians of the Bernoulli era
who was equal to them in genius, but his work
was much less influential.
Today he is most remembered for Taylor Series and the associated
Taylor's Theorem, but he shouldn't get full credit for this.
The method had been anticipated by earlier mathematicians including
Gregory, Leibniz, Newton, and, even earlier, Madhava; and was not fully
appreciated until the work of Maclaurin and Lagrange.

Maclaurin received a University degree in divinity
at age 14, with a treatise on gravitation.
He became one of the most brilliant mathematicians
of his era. He wrote extensively on Newton's method of fluxions,
and the theory of equations, advancing these fields;
worked in optics, and other areas of
mathematical physics; but is most noted
for his work in geometry. Lagrange said Maclaurin's geometry was as beautiful
and ingenious as anything by Archimedes. Clairaut, seeing Maclaurin's
methods, decided that he too would prove theorems with geometry
rather than analysis.
Maclaurin did important work on ellipsoids; for his work on tides he shared the
Paris Prize with Euler and Daniel Bernoulli.
As Scotland's top genius, he was called upon for practical work,
including politics.
Although Maclaurin's work was quite influential, his influence
didn't really match his outstanding brilliance:
he failed to adopt Leibnizian calculus with which great progress
was being made on the Continent, and much of his best work
was published posthumously.
Many of his famous results duplicated work by others:
Maclaurin's Series was just a form of Taylor's series; the Euler-Maclaurin
Summation Formula was also discovered by Euler;
and he discovered the Newton-Cotes Integration Formula after Cotes did.
His brilliant results in geometry included the construction
of a conic from five points, but Braikenridge made the same discovery and
published before Maclaurin did.
He discovered the Maclaurin-Cauchy Test for Integral Convergence
before Cauchy did.
He was first to discover Cramer's Paradox, as Cramer himself acknowledged.
Colin Maclaurin found a simpler and more powerful proof of the
fact that the cycloid solves the famous brachistochrone problem.

Johann Bernoulli had a nephew, three sons and some grandsons who were
all also outstanding mathematicians.
Of these, the most important was his 2nd-oldest son Daniel.
Johann insisted that Daniel study biology and medicine rather than
mathematics, so Daniel specialized initially in
mathematical biology.
He went on to win the Grand Prize of the Paris Academy no less
than ten times, and was a close friend of Euler.
Daniel developed partial differential equations,
preceded Fourier in the use of Fourier series,
did important work in statistics and the theory of equations,
discovered and proved a key theorem about trochoids,
developed a theory of economic risk (motivated by the
St. Petersburg Paradox discovered by his cousin Nicholas),
but is most famous for his key discoveries in mathematical physics:
e.g. the Bernoulli Principle underlying airflight, and
the notion that heat is simply molecules' random kinetic energy.
Daniel Bernoulli is sometimes called the "Founder of Mathematical Physics."

Euler may be the most influential mathematician
who ever lived (though some would make him second to Euclid);
he ranks #77 on Michael Hart's famous list of
the Most Influential Persons in History.
His colleagues called him "Analysis Incarnate."
Laplace, famous for denying credit to fellow mathematicians,
once said "Read Euler: he is our master in everything."
His notations and methods in many areas are in use to this day.
Euler was the most prolific mathematician in history
and is often judged to be the best algorist of all time.
(This brief summary can only touch on a few highlights of Euler's work.
The ranking #4 may seem too low for this supreme mathematician,
but Gauss succeeded at proving several theorems which had stumped Euler.)

Just as Archimedes extended Euclid's geometry to marvelous heights, so
Euler took marvelous advantage of the analysis
of Newton and Leibniz.
He also gave the world modern trigonometry;
pioneered (along with Lagrange) the calculus of variations;
generalized and proved the Newton-Giraud formulae;
and made important contributions to algebra,
e.g. his study of hypergeometric series.
He was also supreme at discrete mathematics,
inventing graph theory.
He also invented the concept of generating functions; for example,
letting p(n) denote the number of partitions of n, Euler found
the lovely equation:
Σnp(n) xn
= 1 / Πk (1 - xk)The denominator of the right side here
expands to a series whose exponents all have the (3m2+m)/2
"pentagonal number" form; Euler found an ingenious proof of this.
Euler wrote the first definitive treatise on continued fractions,
establishing several key theorems on that important topic.

Euler was a very major figure in number theory: He proved that the
sum of the reciprocals of primes less than x is approx. (ln ln x),
invented the totient function and used it to generalize
Fermat's Little Theorem,
found both the largest then-known prime
and the largest then-known perfect number,
proved e to be irrational,
discovered (though without complete proof) a
broad class of transcendental numbers,
proved that all even perfect numbers
must have the Mersenne number form that Euclid had
discovered 2000 years earlier, and much more.
Euler was also first to prove several interesting theorems
of geometry, including facts about the 9-point Feuerbach circle;
relationships among a triangle's altitudes, medians, and
circumscribing and inscribing circles;
the famous Intersecting Chords Theorem;
and an expression for a tetrahedron's volume in terms of its edge lengths.
Euler was first to explore topology, proving theorems
about the Euler characteristic, and the famous
Euler's Polyhedral Theorem, F+V = E+2 (although it may have
been discovered by Descartes and first proved rigorously by Jordan).
Although noted as the first great "pure mathematician,"
Euler's pump and turbine equations revolutionized the design of pumps;
he also made important contributions to music theory,
acoustics, optics, celestial motions, fluid dynamics, and mechanics.
He extended Newton's Laws of Motion to rotating rigid bodies;
and developed the Euler-Bernoulli beam equation.
On a lighter note, Euler constructed a particularly
"magical" magic square.

Euler combined his brilliance with phenomenal concentration.
He developed the first method to estimate the Moon's orbit (the three-body
problem which had stumped Newton), and he settled an arithmetic
dispute involving 50 terms in a long convergent series.
Both these feats were accomplished when he was totally blind.
(About this he said "Now I will have less distraction.")
François Arago said that "Euler calculated without apparent effort,
as men breathe, or as eagles sustain themselves in the wind."

Four of the most important constant symbols in mathematics
(π, e,
i = √-1, and γ = 0.57721566...)
were all introduced or popularized by Euler,
along with operators like Σ.
He did important work with
Riemann's zeta functionζ(s) = ∑ k-s
(although it was not then known
by that name); he anticipated the concept of
analytic continuation by showing
ζ(-1) = 1+2+3+4+... = -1/12.
Euler started as a young student of the Bernoulli family,
and was Daniel Bernoulli's roommate in Saint Petersburg,
where Euler was first employed as a teacher of physiology.
But at age twenty-eight, Euler
discovered the striking identity
ζ(2) = π2/6
This catapulted Euler to instant fame, since the
left-side infinite sum (1 + 1/4 + 1/9 + 1/16 + ...)
was a famous problem of the time.
Euler and others developed alternate proofs and generalizations
of this "Basel problem," and of course the ζ (zeta) function
is now very famous.
Here
is an elegant geometric proof for this theorem.
Among many other famous and important identities,
Euler proved the Pentagonal Number Theorem alluded to above
(a beautiful result which
has inspired a variety of discoveries), and the Euler Product Formula
ζ(s) = ∏(1-p-s)-1
where the right-side product is taken over all primes p.
His most famous identity (which Richard Feynman
called an "almost astounding ... jewel")
unifies the trigonometric and exponential functions:
ei x = cos x + i sin x.
(It is almost wondrous how the particular instance
ei π+1 = 0
combines the most important constants and operators together.)

The reputations of Euler and the Bernoullis
are so high that it is easy to overlook that others
in that epoch made essential contributions to mathematical physics.
(Euler made errors in his development of physics, in some cases because
of a Europeanist rejection of Newton's theories in favor of
the contradictory theories of Descartes and Leibniz.)
The Frenchmen Clairaut and d'Alembert were two other great
and influential mathematicians of the mid-18th century.

Alexis Clairaut was extremely precocious, delivering a
math paper at age 13, and becoming the youngest person ever
elected to the Paris Academy of Sciences.
He developed the concept of skew curves (the earliest precursor
of spatial curvature);
he made very significant contributions in differential
equations and mathematical physics.
Clairaut supported Newton against the Continental schools, and helped
translate Newton's work into French.
The theories of Newton and Descartes gave different predictions
for the shape of the Earth (whether the poles were flattened or
pointy); Clairaut participated in Maupertuis' expedition to Lappland to
measure the polar regions.
Measurements at high latitudes showed the poles to be flattened:
Newton was right.
Clairaut worked on the theories of ellipsoids and the three-body
problem, e.g. Moon's orbit.
That orbit was the major mathematical challenge of the day,
and there was great difficulty reconciling theory and observation.
It was Clairaut who finally resolved this,
by approaching the problem with more rigor than others.
When Euler finally understood Clairaut's solution he called it
"the most important and profound discovery that has ever
been made in mathematics."
Later, when Halley's Comet reappeared as he had predicted,
Clairaut was acclaimed as "the new Thales."

During the century after Newton, the Laws of Motion needed to be
clarified and augmented with mathematical techniques.
Jean le Rond, named after the Parisian church where he
was abandoned as a baby, played a very key role in that development.
His D'Alembert's Principle clarified Newton's Third
Law and allowed problems in dynamics to be expressed with
simple partial differential equations;
his Method of Characteristics then reduced those equations
to ordinary differential equations;
to solve the resultant linear systems,
he effectively invented the method of eigenvalues;
he also anticipated the Cauchy-Riemann Equations.
These are the same techniques in use for many problems
in physics to this day.
D'Alembert was also a forerunner in functions of a complex variable,
and the notions of infinitesimals and limits.
With his treatises on dynamics, elastic collisions,
hydrodynamics, cause of winds, vibrating strings,
celestial motions, refraction, etc., the
young Jean le Rond easily surpassed the efforts of his older
rival, Daniel Bernoulli.
He may have been first to speak of time as a "fourth dimension."
(Rivalry with the Swiss mathematicians led to d'Alembert's
sometimes being unfairly ridiculed, although it does seem true that
d'Alembert had very incorrect notions of probability.)

D'Alembert was first to prove that every
polynomial has a complex root; this is now called the
Fundamental Theorem of Algebra.
(In France this Theorem is called the D'Alembert-Gauss Theorem.
Although Gauss was first to provide a fully rigorous proof,
d'Alembert's proof preceded, and was more nearly complete
than, the attempted proof by Euler-Lagrange.)
He also did creative
work in geometry (e.g. anticipating Monge's Three Circle Theorem),
and was principal creator of the major encyclopedia of his day.
D'Alembert wrote "The imagination in a mathematician who creates
makes no less difference than in a poet who invents."

Lambert had to drop out of school at age 12
to help support his family, but
went on to become a mathematician of great fame and breadth.
He made key discoveries involving continued fractions that
led him to prove that π is irrational.
(He proved more strongly
that tan x and ex are both
irrational for any non-zero rational x.
His proof for this was so remarkable for its time, that its
completeness wasn't recognized for over a century.)
He also conjectured that
π and e were transcendental.
He made advances in analysis (including the
introduction of Lambert's W function)
and in trigonometry (introducing
the hyperbolic functions sinh and cosh);
proved a key theorem of spherical trigonometry,
and solved the "trinomial equation."
Lambert, whom Kant called "the greatest genius of Germany,"
was an outstanding polymath: In addition to several areas of mathematics,
he made contributions in philosophy, psychology,
cosmology (conceiving of star clusters, galaxies and supergalaxies),
map-making (inventing several distinct map projections),
inventions (he built the first practical hygrometer and photometer),
dynamics, and especially optics (several laws of optics carry his name).

Lambert is famous for his work in geometry,
proving Lambert's Theorem (the path of rotation of
a parabola tangent triangle passes through the parabola's focus).
Lagrange declared this famous identity, used to calculate cometary orbits,
to be the most beautiful and significant result in celestial motions.
Lambert was first to explore straight-edge constructions without compass.
He also developed non-Euclidean geometry, long before
Bolyai and Lobachevsky did.

Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia)
was a brilliant man who advanced to
become a teen-age Professor shortly after first studying mathematics.
He excelled in all fields of analysis and number theory;
he made key contributions to the theories of
determinants, continued fractions, and many other fields.
He developed partial differential equations far beyond those
of D. Bernoulli and d'Alembert,
developed the calculus of variations far beyond that of the Bernoullis,
discovered the Laplace transform before Laplace did,
and developed terminology and
notation (e.g. the use of f'(x) and f''(x)
for a function's 1st and 2nd derivatives).
He proved a fundamental Theorem of Group Theory.
He laid the foundations for the theory of
polynomial equations which Cauchy, Abel,
Galois and Poincaré would later complete.
Number theory was almost just a diversion for Lagrange,
whose focus was analysis;
nevertheless he was the master of that field as well,
proving difficult and historic theorems including
Wilson's Theorem (p divides (p-1)! + 1
when p is prime);
Lagrange's Four-Square Theorem (every positive integer is the
sum of four squares);
and that n·x2 + 1 = y2
has solutions for every positive non-square integer n.

Lagrange's many contributions to physics
include understanding of vibrations (he found
an error in Newton's work and published the
definitive treatise on sound), celestial mechanics
(including an explanation of why the Moon keeps the same
face pointed towards the Earth),
the Principle of Least Action
(which Hamilton compared to poetry), and the discovery of
the Lagrangian points (e.g., in Jupiter's orbit).
Lagrange's textbooks were noted for clarity and inspired
most of the 19th-century mathematicians on this list.
Unlike Newton, who used calculus to derive his results but then worked
backwards to create geometric proofs for publication, Lagrange
relied only on analysis.
"No diagrams will be found in this work" he wrote in the preface to his
masterpiece Mécanique analytique.

Lagrange once wrote "As long as algebra and geometry have been
separated, their progress have been slow and their uses limited;
but when these two sciences have been united, they have
lent each mutual forces, and have marched together towards perfection."
Both W.W.R. Ball and E.T. Bell, renowned mathematical historians,
bypass Euler to name Lagrange as "the Greatest
Mathematician of the 18th Century."
Jacobi bypassed Newton and Gauss to call Lagrange "perhaps the greatest
mathematical genius since Archimedes."

Gaspard Monge, son of a humble peddler,
was an industrious and creative inventor
who astounded early with his genius, becoming a professor of
physics at age 16.
As a military engineer he developed the new field of descriptive geometry,
so useful to engineering that it was kept a military secret for 15 years.
Monge made early discoveries in chemistry and helped promote
Lavoisier's work;
he also wrote papers on optics and metallurgy;
Monge's talents were so diverse that he became Minister of the Navy
in the revolutionary government, and eventually became a close friend
and companion of Napoleon Bonaparte.
Traveling with Napoleon he demonstrated great courage
on several occasions.

In mathematics, Monge is called the "Father of Differential Geometry,"
and it is that foundational work for which he is most praised.
He also did work in discrete math, partial differential equations,
and calculus of variations.
He anticipated Poncelet's Principle of Continuity.
Monge's most famous theorems of geometry are the Three Circles Theorem
and Four Spheres Theorem.
His early work in descriptive geometry has little interest to pure
mathematics, but his application of calculus to the
curvature of surfaces inspired Gauss and eventually Riemann, and led
the great Lagrange to say "With [Monge's] application
of analysis to geometry this devil of a man will make
himself immortal."

Monge was an inspirational teacher whose students included
Fourier, Chasles, Brianchon, Ampere, Carnot,
Poncelet, several other famous mathematicians, and perhaps
indirectly, Sophie Germain.
Chasles reports that Monge never drew figures in his lectures,
but could make "the most complicated forms appear in space ...
with no other aid than his hands, whose movements admirably supplemented
his words."
The contributions of Poncelet to synthetic geometry
may be more important than those
of Monge, but Monge demonstrated great genius as an untutored child,
while Poncelet's skills probably developed due to his great teacher.

Laplace was the preeminent mathematical astronomer,
and is often called the "French Newton."
His masterpiece was Mecanique Celeste
which redeveloped and improved Newton's work on planetary motions
using calculus.
While Newton had shown that the two-body gravitation problem
led to orbits which were ellipses (or other conic sections),
Laplace was more interested in the much more difficult problems
involving three or more bodies. (Would Jupiter's pull on Saturn
eventually propel Saturn into a closer orbit, or was
Saturn's orbit stable for eternity?)
Laplace's equations had the optimistic outcome that the solar
system was stable.

Laplace advanced the nebular hypothesis of solar system origin, and
was first to conceive of black holes.
(He also conceived of multiple galaxies,
but this was Lambert's idea first.)
He explained the so-called secular acceleration of the Moon.
(Today we know Laplace's theories do not
fully explain the Moon's path, nor guarantee orbit stability.)
His other accomplishments in physics include theories about
the speed of sound and surface tension.
He worked closely with Lavoisier, helping to discover the elemental
composition of water, and the natures
of combustion, respiration and heat itself.
Laplace may have been first to note that the laws of mechanics
are the same with time's arrow reversed.
He was noted for his strong belief in determinism, famously replying to
Napoleon's question about God with: "I have no need of that hypothesis."

Laplace viewed mathematics as just a tool for developing his
physical theories.
Nevertheless, he made many important mathematical discoveries
and inventions (although the Laplace Transform itself
was already known to Lagrange).
He was the premier expert at differential and difference equations,
and definite integrals.
He developed spherical harmonics, potential theory, and the
theory of determinants; anticipated Fourier's series;
and advanced Euler's technique of generating functions.
In the fields of probability and statistics he made key advances:
he proved the Law of Least Squares,
and introduced the controversial ("Bayesian") rule of succession.
In the theory of equations, he was first to prove that
any polynomial of even degree must have a real quadratic factor.

Others might place Laplace higher on the List,
but he proved no fundamental theorems of pure mathematics
(though his partial differential equation for fluid dynamics
is one of the most famous in physics),
founded no major branch of pure mathematics,
and wasn't particularly concerned with rigorous proof.
(He is famous for skipping difficult proof steps with the
phrase "It is easy to see".)
Nevertheless he was surely one of the
greatest applied mathematicians ever.

Legendre was an outstanding mathematician who
did important work in plane and solid geometry,
spherical trigonometry,
celestial mechanics and other areas of physics,
and especially elliptic integrals and number theory.
He found key results in the theories of sums of squares and
sums of k-gonal numbers.
(For example, he showed that all integers except 4k(8m+7)
can be expressed as the sum of three squares.)
He also made key contributions in several areas of analysis:
he invented the Legendre transform and Legendre polynomials;
the notation for partial derivatives is due to him.
He invented the Legendre symbol; invented the study of zonal harmonics;
proved that π2 was
irrational (the irrationality of π
had already been proved by Lambert);
and wrote important textbooks in several fields.
Although he never accepted non-Euclidean geometry,
and had spent much time trying to prove the Parallel Postulate,
his inspiring geometry text remained a standard until the 20th century.
As one of France's premier mathematicians, Legendre did other
significant work, promoting the careers of Lagrange and Laplace,
developing trig tables, geodesic projects, etc.

There are several important theorems proposed by Legendre
for which he is denied credit, either because his proof was
incomplete or was preceded by another's.
He proposed the famous theorem about primes in a
progression which was proved by Dirichlet; proved and used the
Law of Least Squares which Gauss had left unpublished;
proved the N=5 case of Fermat's Last Theorem which is credited
to Dirichlet; proposed the famous Prime Number Theorem
which was finally proved by Hadamard;
improved the Fermat-Cauchy result about
sums of k-gonal numbers but this topic wasn't fruitful;
and developed various techniques commonly credited to Laplace.
His two most famous theorems of number theory,
the Law of Quadratic Reciprocity
and the Three Squares Theorem (a difficult extension of Lagrange's
Four Squares Theorem), were each enhanced by Gauss a few years
after Legendre's work.
Legendre also proved an early version of Bonnet's Theorem.
Legendre's work in the theory of equations and
elliptic integrals directly inspired the achievements of Galois and Abel
(which then obsoleted much of Legendre's own work);
Chebyshev's work also built on Legendre's foundations.

Joseph Fourier had a varied career:
precocious but mischievous orphan,
theology student, young professor of mathematics
(advancing the theory of equations), then revolutionary activist.
Under Napoleon he was a brilliant and important teacher
and historian; accompanied the French Emperor to Egypt;
and did excellent service as district governor of Grenoble.
In his spare time at Grenoble he continued the work
in mathematics and physics that led to his immortality.
After the fall of Napoleon, Fourier exiled himself to England,
but returned to France when offered an important academic
position and published his revolutionary treatise on the
Theory of Heat.
Fourier anticipated linear programming, developing
the simplex method and Fourier-Motzkin Elimination;
and did significant work in operator theory.
He is also noted for the notion of dimensional analysis,
was first to describe the Greenhouse Effect, and continued
his earlier brilliant work with equations.

Fourier's greatest fame rests on
his use of trigonometric series (now called Fourier series)
in the solution of differential equations.
Since "Fourier" analysis is in extremely common use among
applied mathematicians, he joins the select company of the
eponyms of "Cartesian" coordinates, "Gaussian" curve, and
"Boolean" algebra.
Because of the importance of Fourier analysis,
many listmakers would rank Fourier much higher than I have done;
however the work was not exceptional as pure mathematics.
Fourier's Heat Equation built on Newton's Law of Cooling;
and the Fourier series solution itself
had already been introduced by Euler, Lagrange and Daniel Bernoulli.

Fourier's solution to the heat equation was counterintuitive
(heat transfer doesn't seem to involve the oscillations fundamental
to trigonometric functions): The brilliance of Fourier's imagination
is indicated in that the solution had been rejected
by Lagrange himself.
Although rigorous Fourier Theorems were finally proved
only by Dirichlet, Riemann and Lebesgue,
it has been said that it was Fourier's
"very disregard for rigor" that led to his great achievement,
which Lord Kelvin compared to poetry.

Carl Friedrich Gauss, the "Prince of Mathematics," exhibited
his calculative powers when he corrected his father's arithmetic before
the age of three. His revolutionary nature was demonstrated at age twelve,
when he began questioning the axioms of Euclid. His genius was confirmed
at the age of nineteen when he proved that the regular n-gon was constructible
if and only if it is the product of distinct prime Fermat numbers.
(He didn't complete the proof of the only-if part.
Click
to see construction of regular 17-gon.)
Also at age 19, he proved Fermat's conjecture that every number is
the sum of three triangle numbers.
(He further determined the number of distinct ways such a sum could be formed.)
At age 24 he published Disquisitiones Arithmeticae, probably the
greatest book of pure mathematics ever.

Although he published fewer papers than some other
great mathematicians, Gauss may be the greatest theorem prover ever.
Several important theorems and lemmas bear his name;
his proof of Euclid's Fundamental Theorem of Arithmetic
(Unique Prime Factorization) is considered the first rigorous proof;
he extended this Theorem to the Gaussian (complex) integers;
and he was first to produce a rigorous
proof of the Fundamental Theorem of Algebra
(that an n-th degree polynomial has n complex roots);
his Theorema Egregium ("Remarkable Theorem") that
a surface's essential curvature
derived from its 2-D geometry laid the foundation of differential geometry.
Gauss himself used "Fundamental Theorem" to refer to
Euler's Law of Quadratic Reciprocity; Gauss was first to provide
a proof for this, and provided eight distinct proofs for it
over the years.
Gauss proved the n=3 case of Fermat's Last Theorem for Eisenstein
integers (the triangular lattice-points on the complex plane);
though more general, Gauss' proof was simpler than the real integer proof;
this simplification method revolutionized algebra.
He also found a simpler proof for Fermat's Christmas Theorem, by taking
advantage of the identity x2+y2 = (x + iy)(x - iy).
Other work by Gauss led to fundamental theorems in statistics, vector
analysis, function theory, and generalizations of
the Fundamental Theorem of Calculus.

Gauss built the theory of complex numbers into its modern form, including
the notion of "monogenic" functions
which are now ubiquitous in mathematical physics.
(Constructing the regular 17-gon as a teenager was actually an
exercise in complex-number algebra, not geometry.)
Gauss developed the arithmetic of congruences and became
the premier number theoretician of all time.
Other contributions of Gauss include
hypergeometric series,
foundations of statistics, and differential geometry.
He also did important work in geometry,
providing an improved solution to Apollonius' famous problem
of tangent circles, stating and
proving the Fundamental Theorem of Normal Axonometry,
and solving astronomical problems related to comet
orbits and navigation by the stars.
Ceres, the first asteroid, was discovered when Gauss was a young man;
but only a few observations were made before it disappeared into the
Sun's brightness. Could its orbit be predicted well enough
to rediscover it on re-emergence? Laplace, one of the most
respected mathematicians of the time, declared it impossible.
Gauss became famous when he used an 8th-degree polynomial equation
to successfully predict Ceres' orbit.
Gauss also did important work in several areas of physics,
developed an important modification to Mercator's map projection,
invented the heliotrope, and co-invented the telegraph.

Much of Gauss's work wasn't published: unbeknownst to his colleagues it was
Gauss who first discovered non-Euclidean geometry (even
anticipating Einstein by suggesting physical space
might not be Euclidean),
doubly periodic elliptic functions,
a prime distribution formula,
quaternions, foundations of topology, the Law of Least Squares,
Dirichlet's class number formula,
the key Bonnet's Theorem of differential geometry
(now usually called Gauss-Bonnet Theorem),
the butterfly procedure for rapid calculation of Fourier series,
and even the rudiments of knot theory.
Gauss was first to prove the Fundamental Theorem of Functions of
a Complex Variable (that the line-integral over a closed
curve of a monogenic function is zero),
but he let Cauchy take the credit.
Gauss was very prolific, and may be the most
brilliant theorem prover who ever lived, so many would rank him #1.
But several others on the list had more historical importance.
Abel hints at a reason for this:
"[Gauss] is like the fox, who effaces his tracks in the sand."

Gauss once wrote "It is not knowledge, but the act of learning, ...
which grants the greatest enjoyment.
When I have clarified and exhausted a subject,
then I turn away from it, in order to go into darkness again ..."

Siméon Poisson was a protégé of Laplace and,
like his mentor, is
among the greatest applied mathematicians ever.
Poisson was an extremely prolific researcher and also an excellent
teacher.
In addition to important advances in several areas of physics,
Poisson made key contributions to Fourier analysis, definite
integrals, path integrals, statistics, partial differential equations,
calculus of variations and other fields of mathematics.
Dozens of discoveries are named after Poisson; for example
the Poisson summation formula which has applications in analysis,
number theory, lattice theory, etc.
He was first to note the paradoxical properties of the Cauchy distribution.
He made improvements to Lagrange's equations of celestial
motions, which Lagrange himself found inspirational.
Another of Poisson's contributions to mathematical
physics was his conclusion that the wave theory of light
implies a bright Arago spot at the center of certain shadows.
(Poisson used this paradoxical result to argue that
the wave theory was false, but instead the Arago spot,
hitherto hardly noticed, was observed experimentally.)
Poisson once said "Life is good for only two things,
discovering mathematics and teaching mathematics."

Bolzano was an ordained Catholic priest, a religious philosopher,
and focused his mathematical attention on fields like metalogic,
writing "I prized only ... mathematics which was ... philosophy."
Still he made several important mathematical discoveries ahead of his time.
His liberal religious philosophy upset the Imperial rulers; he
was charged with heresy, placed under house arrest, and his
writings censored. This censorship meant that
many of his great discoveries turned up
only posthumously, and were first rediscovered by others.
He was noted for advocating great rigor, and is appreciated
for developing the (ε, δ) approach for
rigorous proofs in analysis; this work inspired the great Weierstrass.

Bolzano gave the first analytic proof of the Fundamental Theorem
of Algebra; the first rigorous proof that continuous functions
achieve any intermediate value (Bolzano's Theorem, rediscovered
by Cauchy);
the first proof that a bounded sequence of reals has a convergent
subsequence (Bolzano-Weierstrass theorem);
was first to describe a nowhere-differentiable continuous function;
and anticipated Cantor's discovery of the distinction
between denumerable and non-denumerable infinities.
If he had focused on mathematics and published more, he might be
considered one of the most important mathematicians of his era.

After studying under Monge, Poncelet became an
officer in Napoleon's army, then a prisoner of the Russians.
To keep up his spirits as a prisoner he devised and solved
mathematical problems using charcoal and the walls of his
prison cell instead of pencil and paper.
During this time he reinvented projective geometry.
Regaining his freedom, he wrote many papers, made numerous
contributions to geometry;
he also made contributions to practical mechanics.
Poncelet is considered one of the most influential
geometers ever; he is especially noted for his
Principle of Continuity, an intuition with broad application.
His notion of imaginary solutions in geometry was inspirational.
Although projective geometry had been studied earlier
by mathematicians like Desargues, Poncelet's work
excelled and served as an inspiration for other
branches of mathematics including algebra,
topology, Cayley's invariant
theory and group-theoretic developments by Lie and Klein.
His theorems of geometry include his Closure Theorem
about Poncelet Traverses, the Poncelet-Brianchon
Hyperbola Theorem, and Poncelet's Porism (if two conic
sections are respectively inscribed and circumscribed by
an n-gon, then there are infinitely many such n-gons).
Perhaps his most famous theorem, although it was left to
Steiner to complete a proof, is the beautiful Poncelet-Steiner
Theorem about straight-edge constructions.

Cauchy was extraordinarily prodigious, prolific and inventive.
Home-schooled, he awed famous mathematicians at an early age.
In contrast to Gauss and Newton, he was almost over-eager to publish;
in his day his fame surpassed that of Gauss and has continued to grow.
Cauchy did significant work in analysis, algebra, number theory
and discrete topology.
His most important contributions included convergence criteria for
infinite series, the "theory of substitutions"
(permutation group theory), and especially
his insistence on rigorous proofs.

Cauchy's research also included
differential equations, determinants, and probability.
He invented the calculus of residues, rediscovered Bolzano's Theorem,
and much more.
Although he was one of the first great mathematicians
to focus on abstract mathematics (another was Euler),
he also made important contributions to mathematical physics, e.g. the
theory of elasticity.
Cauchy's theorem of solid geometry is important in rigidity theory;
the Cauchy-Schwarz Inequality has very wide application
(e.g. as the basis for Heisenberg's Uncertainty Principle);
several important lemmas of analysis are due to Cauchy;
the famous Burnside's Counting Theorem was first discovered by Cauchy; etc.
He was first to prove Taylor's Theorem rigorously, and
first to prove Fermat's conjecture that every
positive integer can be expressed
as the sum of k k-gonal numbers for any k.
(Gauss had proved the case k = 3.)

One of the duties of a great mathematician is to nurture
his successors, but Cauchy selfishly dropped the ball
on both of the two greatest young mathematicians of his day,
mislaying key manuscripts of both Abel and Galois.
Cauchy is credited with group theory, yet it was Galois who
invented this first, abstracting it far more than Cauchy did,
some of this in a work which Cauchy "mislaid."
(For this historical miscontribution perhaps Cauchy
should be demoted.)

Möbius worked as a Professor of physics and astronomy,
but his astronomy teachers included Carl Gauss and other brilliant
mathematicians, and Möbius is most noted for his work in mathematics.
He had outstanding intuition and originality, and prepared his
books and papers with great care.
He made important advances in number theory, topology,
and especially projective geometry.
Several inventions are named after him, such as the Möbius
transformation and Möbius net of geometry, and
the Möbius function and Möbius inversion formula
of algebraic number theory.
He is most famous for the Möbius strip; this one-sided
strip was first discovered by Lister, but Möbius went much
further and developed important new insights in topology.

Möbius' greatest contributions were to projective geometry,
where he introduced the use of
homogeneous barycentric coordinates as well as signed angles and lengths.
These revolutionary discoveries inspired Plücker, and were
declared by Gauss to be
"among the most revolutionary intuitions in the history of mathematics."

Lobachevsky is famous for discovering non-Euclidean geometry.
He did not regard this new geometry as simply a theoretical
curiosity, writing "There is no branch of mathematics ... which may
not someday be applied to the phenomena of the real world."
He also worked in several branches of analysis and physics, anticipated the
modern definition of function, and may have been first to explicitly note the
distinction between continuous and differentiable curves.
He also discovered the important Dandelin-Gräffe method of
polynomial roots independently of Dandelin and Gräffe.
(In his lifetime, Lobachevsky was under-appreciated and over-worked;
his duties led him to learn architecture and even some medicine.)

Although Gauss and Bolyai discovered non-Euclidean geometry
independently about the same time as Lobachevsky, it is worth noting
that both of them had strong praise for Lobachevsky's genius.
His particular significance was in daring to reject a 2100-year old axiom;
thus William K. Clifford called Lobachevsky "the Copernicus of Geometry."

Chasles was a very original thinker who developed new
techniques for synthetic geometry. He introduced new notions
like pencil and cross-ratio;
made great progress with the Principle of Duality;
and showed how to combine the power of analysis with the intuitions
of geometry.
He invented a theory of characteristics and used it
to become the Founder of Enumerative Geometry.
He proved a key theorem about solid body kinematics.
His influence was very large; for example Poincaré
(student of Darboux, who in turn was Chasles' student)
often applied Chasles' methods.
Chasles was also a historian of mathematics; for example he noted
that Euclid had anticipated the method of cross-ratios.

Jakob Steiner made many major advances in synthetic geometry, hoping that
classical methods could avoid any need for analysis;
and indeed, like Isaac Newton, he was often able to equal or surpass methods
of analysis or the calculus of variations using just pure geometry;
for example he had pure synthetic proofs for a notable extension to Pascal's Mystic Hexagram,
and a reproof of Salmon's Theorem that cubic surfaces have exactly 27 lines.
(He wrote "Calculating replaces thinking while geometry stimulates it.")
One mathematical historian (Boyer) wrote "Steiner reminds one of Gauss
in that ideas and discoveries thronged
through his mind so rapidly that he could scarcely reduce them to order on paper."
Although the Principle of Duality underlying
projective geometry was already known, he gave it a radically
new and more productive basis, and created a new theory of conics.
His work combined generality, creativity and rigor.

Steiner developed several famous construction methods, e.g.
for a triangle's smallest circumscribing and largest inscribing ellipses,
and for its "Malfatti circles."
Among many famous and important
theorems of classic and projective geometry,
he proved that the
Wallace lines of a triangle lie in a 3-pointed hypocycloid,
developed a formula for the partitioning of space by planes,
a fact about the surface areas of tetrahedra,
and proved several facts about his famous
Steiner's Chain of tangential circles and his famous "Roman surface."
Perhaps his three most famous theorems are
the Poncelet-Steiner Theorem (lengths constructible
with straightedge and compass can be constructed with straightedge
alone as long as the picture plane contains the center
and circumference of some circle), the Double-Element Theorem
about self-homologous elements in projective geometry,
and the Isoperimetric Theorem that among solids of equal
volume the sphere will have minimum area, etc.
(Dirichlet found a flaw in the proof of the Isoperimetric Theorem
which was later corrected by Weierstrass.)
Steiner is often called,
along with Apollonius of Perga (who lived 2000 years earlier),
one of the two greatest pure geometers ever.
(The qualifier "pure" is added to exclude such geniuses
as Archimedes, Newton and Pascal from this comparison.
I've included Steiner for his extreme brilliance and productivity:
several geometers had much more historic influence, and as solely
a geometer he arguably lacked "depth.")

Steiner once wrote:
"For all their wealth of content, ... music, mathematics, and chess
are resplendently useless (applied mathematics is a higher plumbing,
a kind of music for the police band). They are metaphysically trivial,
irresponsible. They refuse to relate outward, to take reality for arbiter.
This is the source of their witchery."

Plücker was one of the most innovative geometers,
inventing line geometry (extending the atoms of geometry
beyond just points), enumerative geometry (which considered
such questions as the number of loops in an algebraic curve),
geometries of more than three dimensions, and generalizations
of projective geometry.
He also gave an improved theoretic basis for the Principle of Duality.
His novel methods and notations were important to the development
of modern analytic geometry, and inspired Cayley, Klein and Lie.
He resolved the famous Cramer-Euler Paradox and the related
Poncelet Paradox by studying the singularities of curves;
Cayley described this work
as "most important ... beyond all comparison in the entire subject
of modern geometry."
In part due to conflict with his more famous rival, Jakob Steiner,
Plücker was under-appreciated in his native Germany,
but achieved fame in France and England.
In addition to his mathematical work in algebraic and
analytic geometry,
Plücker did significant work in physics, e.g.
his work with cathode rays.
Although less brilliant as a theorem prover than
Steiner, Plücker's work, taking
full advantage of analysis and seeking physical applications,
was far more influential.

At an early age, Niels Abel studied the works of the
greatest mathematicians, found flaws in their proofs, and resolved to
reprove some of these theorems rigorously.
He was the first to fully prove the general case of Newton's
Binomial Theorem, one of the most widely applied theorems in mathematics.
Several important theorems of analysis are named after Abel,
including the (deceptively simple) Abel's Theorem of
Convergence (published posthumously).
Along with Galois, Abel is considered one of the two founders of group theory.
Abel also made contributions in algebraic geometry
and the theory of equations.

Inversion
(replacing y = f(x) with x = f-1(y))
is a key idea in mathematics (consider Newton's Fundamental Theorem
of Calculus); Abel developed this insight.
Legendre had spent much of his life studying elliptic integrals,
but Abel inverted these to get elliptic functions,
and was first to observe (but in a manuscript mislaid by
Cauchy) that they were doubly periodic.
Elliptic functions quickly became a productive field of mathematics,
and led to more general complex-variable functions,
which were important to the development of
both abstract and applied mathematics.

Finding the roots of polynomials is a key mathematical
problem: the general solution of the quadratic equation was
known by ancients; the discovery of general methods for
solving polynomials of degree three and four is usually treated
as the major math achievement of the 16th century; so
for over two centuries
an algebraic solution for the general 5th-degree polynomial
(quintic) was a Holy Grail
sought by most of the greatest mathematicians.
Abel proved that most quintics did not have such solutions.
This discovery, at the age of only nineteen, would have quickly
awed the world, but Abel was impoverished, had few contacts,
and spoke no German.
When Gauss received Abel's manuscript he discarded it
unread, assuming the unfamiliar author was just another crackpot trying to
square the circle or some such.
His genius was too great for him to be ignored long,
but, still impoverished,
Abel died of tuberculosis at the age of twenty-six.
His fame lives on and even the lower-case word
'abelian' is applied to several concepts.
Liouville said Abel was the greatest genius he ever met.
Hermite said "Abel has left mathematicians enough to keep them busy
for 500 years."

Jacobi was a prolific mathematician who
did decisive work in the algebra and analysis of complex variables,
and did work in number theory
(e.g. cubic reciprocity) which excited Carl Gauss.
He is sometimes described as the successor to Gauss.
As an algorist (manipulator of involved algebraic expressions),
he may have been surpassed only by Euler and Ramanujan.
He was also a very highly regarded teacher.
In mathematical physics, Jacobi perfected Hamilton's
principle of stationary action, and made other important advances.

Jacobi's most significant early achievement was the theory
of elliptic functions, e.g. his fundamental result about
functions with multiple periods.
Jacobi was the first to apply elliptic functions to number theory,
extending Lagrange's famous Four-Squares Theorem to show
in how many distinct ways
a given integer can be expressed as the sum of four squares.
He also made important discoveries in many other areas including
theta functions (e.g. his Jacobi Triple Product Identity),
higher fields, number theory, algebraic geometry,
differential equations, q-series, hypergeometric series,
determinants, Abelian functions, and dynamics.
He devised the algorithms still used to calculate eigenvectors
and for other important matrix manipulations.
The range of his work is suggested by the fact that the
"Hungarian method," an efficient solution to an optimization
problem published more than a century after Jacobi's death,
has since been found among Jacobi's papers.

Like Abel, as a young man
Jacobi attempted to factor the general quintic equation.
Unlike Abel, he seems never to have considered proving its
impossibility.
This fact is sometimes cited to show that despite Jacobi's
creativity, his ill-fated contemporary was the more brilliant genius.

Dirichlet was preeminent in algebraic and analytic
number theory, but did advanced work in several other fields as well:
He discovered the modern definition of function,
the Voronoi diagram of geometry, and important concepts in
differential equations, topology, and statistics.
His proofs were noted both for great ingenuity and unprecedented rigor.
As an example of his careful rigor, he found a fundamental flaw
in Steiner's Isoperimetric Theorem proof which no one else had noticed.
In addition to his own discoveries, Dirichlet played a key role in
interpreting the work of Gauss,
and was an influential teacher,
mentoring famous mathematicians like
Bernhard Riemann (who considered Dirichlet second only to Gauss
among living mathematicians),
Leopold Kronecker and Gotthold Eisenstein.

As an impoverished lad Dirichlet spent his money on
math textbooks; Gauss' masterwork became his life-long companion.
Fermat and Euler had proved the impossibility of
xk + yk = zk
for k = 4 and k = 3; Dirichlet became
famous by proving impossibility for k = 5 at the age of 20.
Later he proved the case k = 14 and, later still, may have helped
Kummer extend Dirichlet's quadratic fields, leading to proofs of
more cases.
More important than his work with Fermat's Last Theorem
was his Unit Theorem, considered one of the most important
theorems of algebraic number theory.
The Unit Theorem is unusually difficult to prove;
it is said that Dirichlet discovered the proof while listening to music
in the Sistine Chapel.
A key step in the proof uses Dirichlet's Pigeonhole Principle,
a trivial idea but which Dirichlet applied with great ingenuity.

Dirichlet did seminal work in analysis and is
considered the founder of analytic number theory.
He invented a method of L-series to prove the important
theorem (Gauss' conjecture)
that any arithmetic series (without a common factor) has an
infinity of primes.
It was Dirichlet who proved the fundamental Theorem of Fourier
series: that periodic analytic functions
can always be represented as a simple
trigonometric series.
Although he never proved it rigorously, he is especially noted
for the Dirichlet's Principle which posits the existence
of certain solutions in the calculus of variations,
and which Riemann found to be particularly fruitful.
Other fundamental results Dirichlet contributed to analysis and
number theory include
a theorem about Diophantine approximations
and his Class Number Formula.

Hamilton was a childhood prodigy.
Home-schooled and self-taught,
he started as a student of languages and literature,
was influenced by an arithmetic prodigy his own age,
read Euclid, Newton and Lagrange, found an error
by Laplace, and made new discoveries in optics;
all this before the age of seventeen when he first
attended school.
At college he enjoyed unprecedented success
in all fields, but his
undergraduate days were cut short abruptly by his
appointment as Royal Astronomer of Ireland at the age of 22.
He soon began publishing his revolutionary treatises on optics,
in which he developed Hamilton's Principle of Stationary Action.
This Principle refined and corrected the earlier principles of
least action developed by Maupertuis, Fermat, and Euler;
it (and related principles) are key to much of modern physics.
His early writing also predicted that some crystals would have an hitherto
unknown "conical" refraction mode; this was soon confirmed experimentally.

Hamilton's Principle of Least Action, and its associated equations
and concept of configuration space, led to
a revolution in mathematical physics.
Since Maupertuis had named this Principle a century
earlier, it is possible to underestimate Hamilton's contribution.
However Maupertuis, along with others credited with anticipating
the idea (Fermat, Leibniz, Euler and Lagrange) failed to
state the full Principle correctly.
Rather than minimizing action, physical systems sometimes achieve
a non-minimal but stationary action in configuration space.
(Poisson and d' Alembert had noticed exceptions to Euler-Lagrange
least action, but failed to find Hamilton's solution.
Jacobi also deserves some credit for the Principle, but his
work came after reading Hamilton.)
Because of this Principle, as well as his wave-particle duality
(which would be further developed by Planck
and Einstein), Hamilton
can be considered a major early influence on quantum theory.

Hamilton also made revolutionary contributions
to dynamics, differential equations, the theory of equations,
numerical analysis, fluctuating functions,
and graph theory (he marketed a puzzle based on his Hamiltonian paths).
He invented the ingenious hodograph.
He coined several mathematical terms including
vector, scalar, associative, and tensor.
In addition to his brilliance and creativity, Hamilton was
renowned for thoroughness and produced voluminous writings
on several subjects.

Hamilton himself considered his greatest accomplishment
to be the development of quaternions, a non-Abelian field to
handle 3-D rotations.
While there is no 3-D analog to the
Gaussian complex-number plane
(based on the equation i2 = -1 ),
quaternions derive from a 4-D analog based on
i2 = j2 =
k2 = ijk = -jik = -1.
Although matrix and tensor methods may seem more general,
quaternions are still in wide engineering use because of
practical advantages, e.g. avoidance of "gimbal lock."

Hamilton once wrote:
"On earth there is nothing great but man;
in man there is nothing great but mind."

Grassmann was an exceptional polymath:
the term Grassmann's Law
is applied to two separate facts in the fields of optics
and linguistics, both discovered by Hermann Grassmann.
He also did advanced work in crystallography, electricity, botany,
folklore, and also wrote on political subjects.
He had little formal training in mathematics, yet single-handedly developed
linear algebra, vector and tensor calculus, multi-dimensional geometry,
new results about cubic surfaces,
the theory of extension, and exterior algebra;
most of this work was so innovative it was not
properly appreciated in his own lifetime.
(Heaviside rediscovered vector analysis many years later.)
Grassmann's exterior algebra, and the associated
concept of Grassmannian manifold, provide a simplifying framework
for many algebraic calculations.
Recently their use led to an important simplification
in quantum physics calculations.

Of his linear algebra, one historian wrote "few have come closer than
Hermann Grassmann to creating, single-handedly, a new subject."
Important mathematicians inspired directly by Grassmann
include Peano, Klein, Cartan, Hankel, Clifford, and Whitehead.

Liouville did expert research in several areas
including number theory, differential geometry, complex
analysis (especially Sturm-Liouville theory, boundary value problems
and dynamical analysis),
harmonic functions,
topology and mathematical physics.
Several theorems bear his name, including
the key result that any bounded entire
function must be constant (the Fundamental Theorem of
Algebra is an easy corollary of this!);
important results in differential equations,
differential algebra, differential geometry;
a key result about conformal mappings;
and an invariance law about trajectories in phase space
which leads to the Second Law of Thermodynamics and is
key to Hamilton's work in physics.
He was first to prove the existence of transcendental numbers.
(His proof was constructive, unlike that of Cantor which came 30 years later).
He invented Liouville integrability and fractional calculus;
he found a new proof of the Law of Quadratic Reciprocity.
In addition to multiple Liouville Theorems, there are two
"Liouville Principles": a fundamental result in differential algebra,
and a fruitful theorem in number theory.
Liouville was hugely prolific in number theory
but this work is largely overlooked, e.g.
the following remarkable generalization
of Aryabhata's identity:
for all N,
Σ (da3)
= (Σ da)2where da is the number of divisors of a,
and the sums are taken over all divisors a of N.

Liouville established an important journal;
influenced Catalan, Jordan, Chebyshev, Hermite;
and helped promote other mathematicians' work,
especially that of Évariste Galois, whose important results
were almost unknown until Liouville clarified them.
In 1851 Augustin Cauchy was bypassed
to give a prestigious professorship to Liouville instead.

Despite poverty, Kummer became an important mathematician
at an early age, doing work with hypergeometric series,
functions and equations, and number theory.
He worked on the 4-degree Kummer Surface, an
important algebraic form which inspired Klein's early work.
He solved the ancient problem of finding all rational quadrilaterals.
His most important discovery was ideal numbers;
this led to the theory of ideals and p-adic numbers; this
discovery's revolutionary nature
has been compared to that of non-Euclidean geometry.
Kummer is famous for his attempts to prove, with the aid
of his ideal numbers, Fermat's Last Theorem.
He established that theorem for almost all exponents (including
all less than 100) but not the general case.

Kummer was an inspirational teacher;
his famous students include Cantor, Frobenius, Fuchs, Schwarz,
Gordan, Joachimsthal, Bachmann, and Kronecker.
(Leopold Kronecker was a brilliant genius sometimes ranked ahead of Kummer
in lists like this; that Kummer was Kronecker's teacher at
high school persuades me to give Kummer priority.)

Galois, who died before the age of twenty-one, not only never
became a professor, but was barely allowed to study as
an undergraduate.
His output of papers, mostly published posthumously,
is much smaller than most of the others on this list, yet it is
considered among the most awesome works in mathematics.
He applied group theory to the theory of equations,
revolutionizing both fields.
(Galois coined the mathematical term group.)
While Abel was the first to prove that some polynomial
equations had no algebraic solutions, Galois established
the necessary and sufficient condition for algebraic solutions to exist.
His principal treatise was a letter he wrote
the night before his fatal duel, of which
Hermann Weyl wrote: "This letter, if judged by the novelty and profundity
of ideas it contains, is perhaps the most substantial piece of writing in
the whole literature of mankind."

Galois' ideas were very far-reaching; for example he is
credited as first to prove that trisecting a general angle with
Plato's rules is impossible.
Galois is sometimes cited (instead of Archimedes, Gauss or
Ramanujan) as "the greatest mathematical genius ever."
His last words (spoken to his brother) were
"Ne pleure pas, Alfred!
J'ai besoin de tout mon courage pour mourir à vingt ans!"
This tormented life, with its pointless early end, is one
of the great tragedies of mathematical history.
Although Galois' group theory is considered
one of the greatest developments of 19th century mathematics,
Galois' writings were largely ignored until
the revolutionary work of Klein and Lie.

Sylvester made important contributions in
matrix theory, invariant theory,
number theory, partition theory, reciprocant theory,
geometry, and combinatorics.
He invented the theory of elementary divisors, and co-invented
the law of quadratic forms.
It is said he coined more new mathematical terms (e.g. matrix,
invariant, discriminant, covariant, syzygy,
graph, Jacobian)
than anyone except Leibniz.
Sylvester was especially noted for the broad range of his mathematics
and his ingenious methods.
He solved (or partially solved) a huge variety of rich puzzles
including various geometric gems;
the enumeration of polynomial roots first tackled by
Descartes and Newton; and, by advancing the theory of
partitions, the system of equations
posed by Euler as The Problem of the Virgins.
Sylvester was also a linguist, a poet, and did work in mechanics (inventing
the skew pantograph) and optics.
He once wrote, "May not music be described as the mathematics of
the sense, mathematics as music of the reason?"

Weierstrass devised new definitions for
the primitives of calculus, developed the concept
of uniform convergence, and was
then able to prove several fundamental but hitherto
unproven theorems.
Starting strictly from the integers,
he also applied his axiomatic methods to a definition
of irrational numbers.
He developed important new insights in other fields
including the calculus of variations, elliptic functions, and trigonometry.
Weierstrass shocked his colleagues when he demonstrated
a continuous function which is differentiable nowhere.
(Both this and the Bolzano-Weierstrass Theorem were rediscoveries
of forgotten results by the under-published Bolzano.)
He found simpler proofs of many existing theorems, including
Gauss' Fundamental Theorem of Algebra and the fundamental Hermite-Lindemann
Transcendence Theorem.
Steiner's proof of the Isoperimetric Theorem contained a flaw,
so Weierstrass became the first to supply a fully rigorous
proof of that famous and ancient result.
Peter Dirichlet was a champion of rigor, but Weierstrass
discovered a flaw in the argument for
Dirichlet's Principle of of variational calculus.

Weierstrass demonstrated extreme brilliance as a youth,
but during his college years he detoured into drinking and dueling
and ended up as a degreeless secondary school teacher.
During this time he studied Abel's papers, developed results
in elliptic and Abelian functions, proved
the Laurent expansion theorem before Laurent did, and
independently proved the Fundamental Theorem of Functions of
a Complex Variable.
He was interested in power series and felt that others
had overlooked the importance of Abel's Theorem.
Eventually one of his papers was published in a journal;
he was immediately given an honorary doctorate
and was soon regarded as one of the best and most
inspirational mathematicians in the world.
His insistence on absolutely rigorous proofs equaled
or exceeded even that of Cauchy, Abel and Dirichlet.
His students included Kovalevskaya, Frobenius, Mittag-Leffler,
and several other famous mathematicians.
Bell called him "probably the greatest mathematical teacher of all time."
In 1873 Hermite called Weierstrass "the Master of all of us."
Today he is often called the "Father of Modern Analysis."

Weierstrass once wrote:
"A mathematician who is not also
something of a poet will never be a complete mathematician."

George Boole was a precocious child who impressed
by teaching himself classical languages, but was too poor
to attend college and
became an elementary school teacher at age 16.
He gradually developed his math skills; as a young man
he published a paper on the calculus of variations, and soon
became one of the most respected mathematicians in England despite
having no formal training.
He was noted for work in symbolic logic, algebra and analysis,
and also
was apparently the first to discover invariant theory.
When he followed up Augustus de Morgan's earlier work in symbolic
logic, de Morgan insisted that Boole was the true master of that field,
and begged his friend to finally study mathematics at university.
Boole couldn't afford to, and had to be appointed Professor instead!

Although very few recognized its importance at the time,
it is Boole's work in Boolean algebra and symbolic logic
for which he is now remembered; this work inspired
computer scientists like Claude Shannon.
Boole's book An Investigation of the Laws of Thought
prompted Bertrand Russell to label him
the "discoverer of pure mathematics."

Boole once said "No matter how correct a mathematical theorem
may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful."

Pafnuti Chebyshev (Pafnuty Tschebyscheff)
was noted for work in probability, number theory,
approximation theory, integrals, the theory of equations,
and orthogonal polynomials.
His famous theorems cover a diverse range;
they include a new version of the Law of Large Numbers,
first rigorous proof of the Central Limit Theorem,
and an important result in integration of radicals first conjectured by Abel.
He invented the Chebyshev polynomials, which have very wide application;
many other theorems or concepts are also named after him.
He did very important work with prime numbers,
proving that there is always a prime
between any n and 2n,
and working with the zeta function before Riemann did.
He made much progress with the Prime Number Theorem,
proving two distinct forms of that theorem,
each incomplete but in a different way.
Chebyshev was very influential for Russian mathematics,
inspiring Andrei Markov and Aleksandr Lyapunov among others.

Chebyshev was also a premier applied mathematician and a renowned inventor;
his several inventions include the Chebyshev linkage,
a mechanical device to convert rotational motion to straight-line motion.
He once wrote "To isolate mathematics from the practical demands of
the sciences is to invite the sterility of a cow shut away from the bulls."

Cayley was one of the most prolific
mathematicians in history;
a list of the branches of mathematics he pioneered will
seem like an exaggeration.
In addition to being very inventive,
he was an excellent algorist; some considered him
to be the greatest mathematician of the late 19th century
(an era that includes Weierstrass and Poincaré).
Cayley was the essential founder of modern group theory,
matrix algebra, the theory of higher singularities,
and higher-dimensional geometry
(building on Plücker's work and anticipating the ideas of Klein),
as well as the theory of invariants.
Among his many important theorems are
the Cayley-Hamilton Theorem,
and Cayley's Theorem itself
(that any group is isomorphic to a subgroup of a symmetric group).
He extended Hamilton's quaternions and developed the octonions,
but was still one of the first to realize that these special
algebras could often be subsumed by general matrix methods.
(Hamilton's friend John T. Graves independently discovered the
octonions about the same time as Cayley did.)
He also did original research in combinatorics (e.g. enumeration of trees),
elliptic and Abelian functions,
and projective geometry.
One of his famous geometric theorems is a generalization of
Pascal's Mystic Hexagram result; another resulted in an elegant
proof of the Quadratic Reciprocity law.

Cayley may have been the least eccentric
of the great mathematicians:
In addition to
his life-long love of mathematics, he enjoyed hiking,
painting, reading fiction, and had a happy married life.
He easily won Smith's Prize and Senior Wrangler at Cambridge,
but then worked as a lawyer for many years.
He later became professor,
and finished his career in the limelight as
President of the British Association for the Advancement of Science.
He and James Joseph Sylvester
were a source of inspiration to each other.
These two, along with Charles Hermite, are considered the
founders of the important theory of invariants.
Though applied first to algebra, the notion of invariants
is useful in many areas of mathematics.

Cayley once wrote:
"As for everything else, so for a mathematical theory: beauty can be
perceived but not explained."

Hermite studied the works of Lagrange
and Gauss from an early age and soon developed an alternate
proof of Abel's famous quintic impossibility result.
He attended the same college as Galois and
also had trouble passing their examinations, but soon became
highly respected by Europe's best mathematicians for his
significant advances in analytic number theory, elliptic functions,
and quadratic forms.
Along with Cayley and Sylvester,
he founded the important theory of invariants.
Hermite's theory of transformation allowed him to connect
analysis, algebra and number theory in novel ways.
He was a kindly modest man and an inspirational teacher.
Among his students was Poincaré, who said of Hermite, "He never
evokes a concrete image, yet you soon perceive that the more
abstract entities are to him like living creatures....
Methods always seemed to be born in his mind in some mysterious way."
Hermite's other famous students included Darboux, Borel, and
Hadamard who wrote of "how magnificent Hermite's teaching was,
overflowing with enthusiasm for science, which seemed to come to life
in his voice and whose beauty he never failed to communicate to us,
since he felt it so much himself to the very depth of his being."

Although he and Abel had proved that the general quintic lacked
algebraic solutions, Hermite introduced an elliptic analog to the
circular trigonometric functions and used these to provide
a general solution for the quintic equation.
He developed the concept of complex conjugate which is now
ubiquitous in mathematical physics and matrix theory.
He was first to prove that the Stirling
and Euler generalizations of the factorial function are equivalent.
He was first to note remarkable facts about Heegner numbers, e.g.
eπ√163 = 262537412640768743.9999999999992...
(Without computers he was able to calculate this number,
including the twelve 9's to the right of the decimal point.)
Very many elegant concepts and theorems are named after Hermite.
Hermite's most famous result may be his intricate proof
that e (along with a broad class
of related numbers) is transcendental.
(Extending the proof to π was left to Lindemann,
a matter of regret for historians, some of whom who regard Hermite as the
greatest mathematician of his era.)

Eisenstein was born into severe poverty and suffered health
problems throughout his short life, but was still one of the more significant
mathematicians of his era.
Today's mathematicians who study Eisenstein are invariably amazed
by his brilliance and originality.
He made revolutionary advances in number theory,
algebra and analysis, and was also a composer of music.
He anticipated ring theory, developed a new basis for elliptic
functions, studied ternary quadratic forms,
proved several theorems about
cubic and higher-degree reciprocity, discovered the
notion of analytic covariant, and much more.

Eisenstein was a young prodigy; he once wrote
"As a boy of six I could understand the proof of a mathematical theorem
more readily than that meat had to be cut with one's knife, not one's fork."
Despite his early death, he is considered one
of the greatest number theorists ever.
Gauss named Eisenstein, along with Newton and Archimedes, as one of
the three epoch-making mathematicians of history.

Kronecker was a businessman who pursued mathematics mainly
as a hobby, but was still very prolific,
and one of the greatest theorem provers of his era.
He explored a wide variety of mathematics -- number theory,
algebra, analysis, matrixes -- and especially the interconnections between
areas.
Many concepts and theorems are named after Kronecker;
some of his theorems are frequently used as lemmas in algebraic number theory,
ergodic theory, and approximation theory.
He provided key ideas about foundations and continuity
despite that he had philosophic objections to
irrational numbers and infinities.
He also introduced the Theory of Divisors to avoid Dedekind's Ideals;
the importance of this and other work was only realized long after
his death.
Kronecker's philosophy eventually led to the Constructivism
and Intuitionism of Brouwer and Poincaré.

Riemann was a phenomenal genius whose work
was exceptionally deep, creative and rigorous; he made
revolutionary contributions in many areas of pure mathematics,
and also inspired the development of physics.
He had poor physical health and died at an early age, yet is still
considered to be among the most productive mathematicians ever.
He made revolutionary advances in complex analysis, which he connected to
both topology and number theory.
He applied topology to analysis, and analysis to number theory,
making revolutionary contributions to all three fields.
He introduced the Riemann integral which clarified analysis.
He developed the theory of manifolds, a term which he invented.
Manifolds underpin topology.
By imposing metrics on manifolds Riemann invented
differential geometry and took non-Euclidean geometry far
beyond his predecessors.
Riemann's other masterpieces include tensor analysis,
the theory of functions, and a key relationship between
some differential equation solutions and hypergeometric series.
His generalized notions of distance and curvature
described new possibilities for the geometry of space itself.
Several important theorems and concepts are named
after Riemann, e.g. the Riemann-Roch Theorem, a key connection
among topology, complex analysis and algebraic geometry.
He proved Riemann's Rearrangement Theorem, a strong (and paradoxical)
result about conditionally convergent series.
He was also first to prove theorems named after others, e.g. Green's Theorem.
He was so prolific and original that some of his work
went unnoticed (for example, Weierstrass became famous for showing
a nowhere-differentiable continuous function;
later it was found that Riemann had casually mentioned one
in a lecture years earlier).
Like his mathematical peers (Gauss, Archimedes, Newton), Riemann
was intensely interested in physics.
His theory unifying electricity, magnetism and light was
supplanted by Maxwell's theory; however modern physics, beginning
with Einstein's relativity, relies on Riemann's
curvature tensor and other notions of the geometry of space.

Riemann's teacher was Carl Gauss, who helped steer the young
genius towards pure mathematics. Gauss selected "On the hypotheses
that Lie at the Foundations of Geometry" as Riemann's first lecture;
with this famous lecture Riemann went far beyond Gauss' initial effort
in differential geometry, extended it to multiple dimensions, and
introduced the new and important theory of differential manifolds.
Five years later, to celebrate his election to the Berlin Academy,
Riemann presented a lecture "On the Number of Prime Numbers Less
Than a Given Quantity," for which "Number" he presented and
partially proved an exact formula, albeit weirdly complicated.
Numerous papers have been written on the distribution of primes,
but Riemann's contribution is incomparable, despite that
his Berlin Academy lecture was his only paper ever on the topic,
and number theory was far from his specialty.
In the lecture he posed the Hypothesis of Riemann's zeta function,
needed for the missing step in his proof.
This Hypothesis is considered the most important and famous unsolved problem
in mathematics.
(Asked what he would first do, if he were magically awakened after
centuries, David Hilbert replied "I would ask whether
anyone had proved the Riemann Hypothesis.")
ζ(.) was defined for convergent cases in Euler's mini-bio,
which Riemann extended via analytic continuation for all cases.
The Riemann Hypothesis "simply" states that
in all solutions of ζ(s = a+bi) = 0,
either s has real part a=1/2
or imaginary part b=0.

Despite his great creativity (Gauss praised
Riemann's "gloriously fertile originality;" another biographer called
him "one of the most profound and imaginative mathematicians of all time
[and] a great philosopher"),
Riemann once said: "If only I had the theorems!
Then I should find the proofs easily enough."

Henry Smith (born in Ireland) was one of the greatest number theorists,
working especially with elementary divisors; he also advanced
the theory of quadratic forms.
A famous problem of Eisenstein was, given n
and k, in how many different ways can n
be expressed as the sum of k squares?
Smith made great progress on this problem, subsuming special
cases which had earlier been famous theorems.
Although most noted for number theory, he had great breadth.
He did prize-winning work in geometry, discovered the unique normal form
for matrices which now bears his name, anticipated specific fractals
including the Cantor set, the Sierpinski gasket and the Koch snowflake,
and wrote a paper demonstrating the limitations of Riemann integration.

Smith is sometimes called "the mathematician the world forgot."
His paper on integration could have led directly
to measure theory and Lebesgue integration, but was ignored for decades.
The fractals he discovered are named after people who rediscovered them.
The Smith-Minkowski-Siegel mass formula of lattice theory would be
called just the Smith formula, but had to be rediscovered.
And his solution to the Eisenstein five-squares problem, buried
in his voluminous writings on number theory, was ignored:
this "unsolved" problem was featured for a prize which Minkowski won
two decades later!

Henry Smith was an outstanding intellect with a modest
and charming personality.
He was knowledgeable in a broad range of fields unrelated
to mathematics; his University even insisted he run for Parliament.
His love of mathematics didn't depend on utility: he once wrote
"Pure mathematics: may it never be of any use to anyone."

Luigi Cremona made many important advances in analytic,
synthetic and projective geometry, especially in the transformations
of algebraic curves and surfaces.
Working in mathematical physics, he developed the new field of
graphical statics, and used it to reinterpret some of Maxwell's results.
He improved (or found brilliant proofs for) several results of Steiner,
especially in the field of cubic surfaces.
(Some of this work was done in collaboration with Rudolf Sturm.)
He is especially noted for developing the theory of Cremona transformations
which have very wide application.
He found a generalization of Pascal's Mystic Hexagram.
Cremona also played a political role in establishing the modern Italian state
and, as an excellent teacher, helped make Italy a top
center of mathematics.

At the age of 14, Maxwell published a remarkable paper
on the construction of ovals;
these were an independent discovery of the Ovals of Descartes,
but Maxwell allowed more than two foci, had elaborate configurations
(he was drawing the ovals with string and pencil), and identified
errors in Descartes' treatment of them.
His genius was soon renowned throughout Scotland, with the future Lord Kelvin
remarking that Maxwell's "lively imagination started so many hares that
before he had run one down he was off on another."
He did a comprehensive analysis of Saturn's rings; developed
the important kinetic theory of gases; explored elasticity, viscosity,
knot theory, topology, soap bubbles, and more.
He introduced the "Maxwell's Demon" as a thought experiment for thermodynamics;
his paper "On Governors" effectively founded the field of cybernetics;
he advanced the theory of color, and produced the first color photograph.
One Professor said of him, "there is scarcely a single topic that
he touched upon, which he did not change almost beyond recognition."
Maxwell was also a poet.

Maxwell did little of importance in pure mathematics, so
his great creativity in mathematical physics might not seem enough to
qualify him for this list,
although his contribution to the kinetic theory of gases
(which even led to the first estimate of molecular sizes) would
already be enough to make him one of the greatest physicists.
But then, in 1864 James Clerk Maxwell stunned the world by
publishing the equations of electricity and magnetism,
predicting the existence of radio waves and that light itself
is a form of such waves and is thus linked to the electro-magnetic force.
Richard Feynman considered this the most significant event of
the 19th century (though others might give higher billing to
Darwin's theory of evolution).
While Einstein, Newton, and Galileo may be the Top Three,
Maxwell is a strong candidate for "fourth greatest scientist ever."
Recalling Newton's comment about "standing on the shoulders" of
earlier greats, Einstein was asked whose shoulders he
stood on; he didn't name Newton: he said "Maxwell."
Maxwell has been called the "Father of Modern Physics"; he ranks #24 on
Hart's list of the Most Influential Persons in History.

Dedekind was one of the most innovative mathematicians ever;
his clear expositions and rigorous axiomatic methods had great influence.
He made seminal contributions to abstract algebra and algebraic number theory
as well as mathematical foundations.
He was one of the first to pursue Galois Theory, making major advances there
and pioneering in the application of group theory to other branches
of mathematics.
Dedekind also invented a system of fundamental axioms for arithmetic,
worked in probability theory and complex analysis,
and invented prime partitions and modular lattices.
Dedekind may be most famous for his theory of ideals and rings;
Kronecker and Kummer had begun this, but
Dedekind gave it a more abstract and productive basis,
which was developed further by Hilbert, Noether and Weil.
Though the term ring itself was coined by Hilbert,
Dedekind introduced the terms module, field, and ideal.
Dedekind was far ahead of his time, so Noether became famous
as the creator of modern algebra; but she acknowledged her great predecessor,
frequently saying "It is all already in Dedekind."

Dedekind was concerned with rigor, writing
"nothing capable of proof ought to be accepted without proof."
Before him, the real numbers, continuity, and infinity all
lacked rigorous definitions.
The axioms Dedekind invented allow the integers and rational
numbers to be built and his Dedekind Cut then led to a rigorous
and useful definition of the real numbers.
Dedekind was a key mentor for Georg Cantor:
he introduced the notion that a bijection implied equinumerosity,
used this to define infinitude (a set is infinite if
equinumerous with its proper subset),
and was first to prove the Cantor-Bernstein-Schröder Theorem,
though he didn't publish his proof.
(Because he spent his career at a minor university,
and neglected to publish some of his work,
Dedekind's contributions may be underestimated.)

Alfred Clebsch began in mathematical physics,
working in hydrodynamics and elasticity, but went on
to become a pure mathematician of great brilliance and versatility.
He started with novel results in analysis, but went on to
make important advances to the invariant theory of
Cayley and Sylvester (and Salmon and Aronhold),
to the algebraic geometry and elliptic functions of Abel and Jacobi,
and to the enumerative and projective geometries of Plücker.
He was also one of the first to build on Riemann's innovations.
Clebsch developed new notions, e.g. Clebsch-Aronhold symbolic notation
and 'connex';
and proved key theorems about cubic surfaces
(for example, the Sylvester pentahedron conjecture)
and other high-degree curves,
and representations (bijections) between surfaces.
Some of his work, e.g. Clebsch-Gordan coefficients which are important
in physics, was done in collaboration with Paul Gordan.
For a while Clebsch was one of the top mathematicians in Germany,
and founded an important journal, but he died young.
He was a key teacher of Max Noether, Ferdinand Lindemann,
Alexander Brill and Gottlob Frege.
Clebsch's great influence is suggested by the fact that
his name appeared as co-author on a text published 60 years after his death.

Beltrami was an outstanding mathematician noted
for early insights connecting geometry and topology (differential
geometry, pseudospherical surfaces, etc.), transformation theory,
differential calculus, and especially for proving
the equiconsistency of hyperbolic and Euclidean geometry
for every dimensionality;
he achieved this by building on models of Cayley, Klein,
Riemann and Liouville.
He was first to invent singular value decompositions.
(Camille Jordan and J.J. Sylvester each invented it independently
a few years later.)
Using insights from non-Euclidean geometry, he did important
mathematical work in a very wide range of physics;
for example he improved Green's theorem, generalized the Laplace operator,
studied gravitation in non-Euclidean space,
and gave a new derivation of Maxwell's equations.

Jordan was a great "universal mathematician",
making revolutionary advances in group theory, topology, and
operator theory;
and also doing important work in differential equations, number theory,
measure theory,
matrix theory, combinatorics, algebra and especially Galois theory.
He worked as both mechanical engineer and professor of analysis.
Jordan is especially famous for the Jordan Closed Curve Theorem of topology,
a simple statement "obviously true" yet remarkably difficult to prove.
In measure theory he developed Peano-Jordan "content"
and proved the Jordan Decomposition Theorem.
He also proved the Jordan-Holder Theorem of group theory,
invented the notion of homotopy,
invented the Jordan Canonical Forms of matrix theory,
and supplied the first complete proof of
Euler's Polyhedral Theorem, F+V = E+2.
Some consider Jordan second only to Weierstrass among great
19th-century teachers; his work inspired such mathematicians
as Klein, Lie and Borel.

Lie was twenty-five years old before his interest in
and aptitude for mathematics became clear,
but then did revolutionary
work with continuous symmetry and continuous transformation groups.
These groups and the algebra he developed to manipulate
them now bear his name; they have major importance in
the study of differential equations.
Lie sphere geometry is one result of Lie's fertile approach
and even led to a new approach for Apollonius'
ancient problem about tangent circles.
Lie became a close friend and collaborator of Felix Klein early in
their careers; their methods of relating group theory to
geometry were quite similar;
but they eventually fell out after Klein became (unfairly?)
recognized as the superior of the two.
Lie's work wasn't properly appreciated in his own lifetime, but
one later commentator was "overwhelmed by the richness and beauty
of the geometric ideas flowing from Lie's work."

Darboux did outstanding work in geometry,
differential geometry, analysis, function theory,
mathematical physics, and other fields,
his ability "based on a rare combination of
geometrical fancy and analytical power."
He devised the Darboux integral, equivalent to Riemann's integral
but simpler;
developed a novel mapping between (hyper-)sphere
and (hyper-)plane; proved an important Envelope Theorem in the
calculus of variations; developed the field
of infinitesimal geometry; and more.
Several important theorems are named after him including
a generalization of Taylor series, the foundational theorem
of symplectic geometry, and the fact that "the image of an interval
is also an interval."
He wrote the definitive textbook on differential geometry;
he was an excellent teacher, inspiring Borel, Cartan and others.

Clifford was a versatile and talented mathematician who was
among the first to appreciate the work of both Riemann and Grassmann.
He found new connections between algebra, topology and non-Euclidean geometry.
Combining Hamilton's quaternions, Grassmann's exterior algebra,
and his own geometric intuition and understanding of physics,
he developed biquaternions, and generalized this
to geometric algebra, which paralleled work by Klein.
In addition to developing theories, he also produced ingenious proofs;
for example he was first to prove Miquel's n-Circle Theorem,
and did so with a purely geometric argument.
Clifford is especially famous for anticipating, before Einstein,
that gravitation could be modeled with a non-Euclidean space.
He was a polymath; a talented teacher, noted philosopher,
writer of children's fairy tales, and outstanding athlete.
With his singular genius, Clifford
would probably have become one of the greatest mathematicians
of his era had he not died at age thirty-three.

Cantor did brilliant and important work early in his career,
for example
he greatly advanced the Fourier-series uniqueness question which
had intrigued Riemann. In his explorations of that problem he was led
to questions of set enumeration, and his greatest invention: set theory.
Cantor created modern Set Theory almost single-handedly,
defining cardinal numbers, well-ordering, ordinal numbers,
and discovering the Theory of Transfinite Numbers.
He defined equality between cardinal numbers based on the
existence of a bijection, and was the first to demonstrate that
the real numbers have a higher cardinal number than the
integers.
(He also showed that the rationals have the same cardinality
as the integers; and that the reals have the same cardinality
as the points of N-space and as the power-set of the integers.)
Although there are infinitely many distinct transfinite numbers,
Cantor conjectured that C, the cardinality of
the reals, was the second smallest transfinite number.
This Continuum Hypothesis was included in Hilbert's famous
List of Problems, and was partly resolved many years later:
Cantor's Continuum Hypothesis is an "Undecidable Statement"
of Set Theory.

Cantor's revolutionary set theory attracted vehement opposition
from Poincaré ("grave disease"), Kronecker (Cantor was a "charlatan"
and "corrupter of youth"), Wittgenstein ("laughable nonsense"),
and even theologians.
David Hilbert had kinder words for it:
"The finest product of mathematical genius and one of the
supreme achievements of purely intellectual human activity"
and addressed the critics with "no one shall expel us from
the paradise that Cantor has created."
Cantor's own attitude was expressed with
"The essence of mathematics lies in its freedom."
Cantor's set theory laid the theoretical basis for the measure theory
developed by Borel and Lebesgue.
Cantor's invention of modern set theory
is now considered one of the most important and creative achievements
in modern mathematics.

Cantor demonstrated much breadth
(he even involved himself in the Shakespeare authorship controversy!).
In addition to his set theory and key discoveries in the
theory of trigonometric series, he made advances in number theory,
and gave the modern definition of irrational numbers.
His Cantor set was the early inspiration for fractals.
Cantor was also an excellent violinist.
He once wrote
"In mathematics the art of proposing a question must be held
of higher value than solving it."

Gottlob Frege developed the first complete and fully rigorous
system of pure logic;
his work has been called the greatest advance in logic since Aristotle.
He introduced the essential notion of quantifiers; he distinguished
terms from predicates, and simple predicates from 2nd-level predicates.
From his second-order logic he defined numbers, and derived
the axioms of arithmetic with what is now called Frege's Theorem.
His work was largely underappreciated at the time, partly
because of his clumsy notation, partly because his system
was published with a flaw (Russell's antinomy).
(Bertrand Russell reports that when he informed him of this flaw, Frege
took it with incomparable integrity, grace, and even intellectual pleasure.)
Frege and Cantor were the era's outstanding foundational theorists;
unfortunately their relationship with each other became bitter.
Despite all this, Frege's work influenced Peano, Russell, Wittgenstein
and others; and he is now often called the greatest
mathematical logician ever.

Frege also did work in geometry and differential equations;
and, in order to construct the real numbers with his set theory,
proved an important new theorem of group theory.
He was also an important philosopher, and wrote "Every good mathematician
is at least half a philosopher, and every good philosopher is at least
half a mathematician."

Frobenius did significant work in a very broad range of mathematics,
was an outstanding algorist,
and had several successful students including Edmund Landau, Issai Schur,
and Carl Siegel.
In addition to developing the theory of abstract groups,
Frobenius did important work in number theory, differential equations,
elliptic functions, biquadratic forms, matrixes, and algebra.
He was first to actually prove the important Cayley-Hamilton Theorem,
and first to extend the Sylow Theorems to abstract groups.
He anticipated the important and imaginative Prime Density Theorem,
though he didn't prove its general case.
Although he modestly left his name off the "Cayley-Hamilton Theorem,"
many lemmas and concepts are named after him, including
Frobenius conjugacy class, Frobenius reciprocity,
Frobenius manifolds, the Frobenius-Schur Indicator, etc.
He is most noted for his character theory,
a revolutionary advance which led to the representation theory of groups,
and has applications in modern physics.
The middle-aged Frobenius invented this after the aging Dedekind asked him
for help in solving a key algebraic factoring problem.

Klein's key contribution was an application of
invariant theory to unify geometry with group theory.
This radical new view of geometry
inspired Sophus Lie's Lie groups, and also led
to the remarkable unification of Euclidean and non-Euclidean geometries
which is probably Klein's most famous result.
Klein did other work in function theory, providing links between
several areas of
mathematics including number theory, group theory,
hyperbolic geometry, and abstract algebra.
His Klein's Quartic curve and popularly-famous
Klein's bottle were among several useful results
from his new approaches to groups and higher-dimensional geometries
and equations.
Klein did significant work in mathematical physics,
e.g. writing about gyroscopes.
He facilitated David Hilbert's early career, publishing
his controversial Finite Basis Theorem and declaring it
"without doubt the most important work on
general algebra [the leading German journal] ever published."

Klein is also famous for his book on the icosahedron,
reasoning from its symmetries to
develop the elliptic modular and automorphic functions which he
used to solve the general quintic equation.
He formulated a "grand uniformization theorem" about automorphic functions
but suffered a health collapse before completing the proof.
His focus then changed to teaching; he devised a mathematics
curriculum for secondary schools which
had world-wide influence.
Klein once wrote "... mathematics has been most advanced by those who
distinguished themselves by intuition rather than by rigorous proofs."

Heaviside dropped out of high school to teach himself
telegraphy and electromagnetism, becoming first a telegraph
operator but eventually perhaps the greatest electrical
engineer ever.
He developed transmission line theory, invented the coaxial cable,
predicted Cherenkov radiation,
described the use of the ionosphere in radio transmission, and much more.
Some of his insights anticipated parts of special relativity, and
he was first to speculate about gravitational waves.
For his revolutionary discoveries in electromagnetism and mathematics,
Heaviside became the first winner of the Faraday Medal.

As an applied mathematician, Heaviside developed operational
calculus (an important shortcut for solving differential equations);
developed vector analysis independently of Grassmann; and
demonstrated the usage of complex numbers for electro-magnetic equations.
Four of the famous Maxwell's Equations are in fact due to Oliver Heaviside,
Maxwell having presented a more cumbersome version.
Although one of the greatest applied mathematicians, Heaviside
is omitted from the Top 100 because he didn't provide proofs for his methods.
Of this Heaviside said,
"Should I refuse a good dinner simply because I do not
understand the process of digestion?"

Sofia Kovalevskaya (aka Sonya Kowalevski;
née Korvin-Krukovskaya)
was initially self-taught, sought out Weierstrass as her teacher,
and was later considered the greatest female mathematician ever
(before Emmy Noether).
She was influential in the development of Russian mathematics.
Kovalevskaya studied Abelian integrals and partial differential equations,
producing the important Cauchy-Kovalevsky Theorem;
her application of complex analysis to physics inspired Poincaré
and others.
Her most famous work was the solution to the Kovalevskaya top,
which has been called a "genuine highlight of 19th-century mathematics."
Other than the simplest cases solved by Euler and Lagrange,
exact ("integrable") solutions to the
equations of motion were unknown, so Kovalevskaya received fame
and a rich prize when she solved the Kovalevskaya top.
Her ingenious solution might be considered a mere curiosity,
but since it is still the only post-Lagrange physical motion problem for which
an "integrable" solution has been demonstrated, it
remains an important textbook example.
Kovalevskaya once wrote "It is impossible to be a mathematician
without being a poet in soul."
She was also a noted playwright.

Poincaré founded the theory of algebraic (combinatorial)
topology, and is sometimes called the "Father of Topology"
(a title also used for Euler and Brouwer).
He also did brilliant work in several other areas of mathematics;
he was one of the most creative mathematicians ever,
and the greatest mathematician of the Constructivist
("intuitionist") style.
He published hundreds of papers on a variety of topics and might have
become the most prolific mathematician ever,
but he died at the height of his powers.
Poincaré was clumsy and absent-minded; like Galois,
he was almost denied admission to French University,
passing only because at age 17 he was already far too famous to flunk.

In addition to his topology,
Poincaré laid the foundations of homology;
he discovered automorphic functions (a unifying
foundation for the trigonometric and elliptic functions),
and essentially founded the theory of periodic orbits;
he made major advances in the theory of differential equations.
He is credited with partial solution of Hilbert's 22nd Problem.
Several important results carry his name, for example the
famous Poincaré Recurrence Theorem, which almost seems to
contradict the Second Law of Thermodynamics.
Poincaré is especially noted for effectively discovering chaos theory,
and for posing Poincaré's Conjecture;
that conjecture was one of the most famous
unsolved problems in mathematics for an entire century,
and can be explained without equations to a layman.
The Conjecture is that all "simply-connected" closed manifolds
are topologically equivalent to "spheres"; it is directly
relevant to the possible topology of our universe.
Recently Grigori Perelman proved Poincaré's conjecture,
and is eligible for the first Million Dollar
math prize in history.

As were most of the greatest mathematicians,
Poincaré was intensely interested in physics.
He made revolutionary advances in fluid dynamics and
celestial motions; he anticipated Minkowski space
and much of Einstein's Special Theory of Relativity
(including the famous equation E = mc2).
Poincaré also found time to become
a famous popular writer of philosophy, writing,
"Mathematics is the art of giving the same name to different things;"
and "A [worthy] mathematician experiences in his work
the same impression as an artist; his
pleasure is as great and of the same nature;"
and "If nature were not beautiful, it would not be worth knowing,
and if nature were not worth knowing, life would not be worth living."
With his fame, Poincaré helped the world recognize the importance of the
new physical theories of Einstein and Planck.

Markov did excellent work in a broad range of mathematics
including analysis, number theory, algebra,
continued fractions, approximation theory, and especially
probability theory:
it has been said that his accuracy and clarity transformed
probability theory into one of the most perfected areas of mathematics.
Markov is best known as the founder of the theory of stochastic processes.
In addition to his Ergodic Theorem about such processes, theorems
named after him include the Gauss-Markov Theorem of statistics,
the Riesz-Markov Theorem of functional analysis, and
the Markov Brothers' Inequality in the theory of equations.
Markov was also noted for his politics, mocking Czarist rule,
and insisting that he be excommunicated from the Russian Orthodox Church
when Tolstoy was.

Markov had a son, also named Andrei Andreyevich, who was also an
outstanding mathematician of great breadth.
Among the son's achievements was Markov's Theorem, which helps relate
the theories of braids and knots to each other.

Giuseppe Peano is one of the most under-appreciated
of all great mathematicians.
He started his career by proving a fundamental
theorem in differential equations, developed practical
solution methods for such equations, discovered a continuous
space-filling curve (then thought impossible), and laid
the foundations of abstract operator theory.
He also produced the best calculus textbook of his time,
was first to produce a correct (non-paradoxical) definition of
surface area, proved an important
theorem about Dirichlet functions, did important work
in topology, and much more.
Much of his work was unappreciated and left for others to rediscover:
he anticipated many of Borel's and Lebesgue's
results in measure theory, and several concepts and theorems
of analysis.
He was the champion of counter-examples, and found flaws in
published proofs of several important theorems.

Most of the preceding work was done when Peano was quite young.
Later he focused on mathematical foundations, and this is the work
for which he is most famous.
He developed rigorous definitions and axioms for
set theory, as well as most of the notation of modern set theory.
He was first to define arithmetic (and then the rest
of mathematics) in terms of set theory.
Peano was first to note that some proofs required an explicit Axiom of Choice
(although it was Ernst Zermelo who explicitly formulated that Axiom a
few years later).

Despite his early show of genius, Peano's quest for utter
rigor may have detracted from his influence in mainstream mathematics.
Moreover, since he modestly referenced work by predecessors like
Dedekind, Peano's huge influence in axiomatic theory is often overlooked.
Yet Bertrand Russell reports that it was from Peano
that he first learned that a single-member set is not the same
as its element; this fact is now taught in elementary school.

Vito Volterra founded the field of functional analysis
('functions of lines'), and used it to extend the work of
Hamilton and Jacobi to more areas of mathematical physics.
He developed cylindrical waves and the theory of integral equations.
He worked in mechanics, developed the theory of crystal dislocations,
and was first to propose the use of helium in balloons.
Eventually he turned to mathematical biology and made notable
contributions to that field, e.g. predator-prey equations.

Hilbert, often considered the greatest
mathematician of the 20th century,
was unequaled in many fields of mathematics,
including axiomatic theory, invariant theory,
algebraic number theory, class field theory and functional analysis.
He proved many new theorems, including
the fundamental theorems of algebraic manifolds,
and also discovered simpler proofs for older theorems.
His examination of calculus led him
to the invention of Hilbert space,
considered one of the key concepts of functional analysis
and modern mathematical physics.
His Nullstellensatz Theorem laid the foundation of algebraic geometry.
He was a founder of fields like metamathematics and modern logic.
He was also the founder of the "Formalist" school which opposed the
"Intuitionism" of Kronecker and Brouwer.
He developed a new system of definitions and axioms
for geometry, replacing the 2200 year-old system of Euclid.
As a young Professor he proved his Finite Basis Theorem,
now regarded as one of the most important results of
general algebra.
His mentor, Paul Gordan, had sought the proof for many years, and
rejected Hilbert's proof as non-constructive.
Later, Hilbert produced the first constructive proof of the
Finite Basis Theorem, as well.
In number theory, he proved Waring's famous conjecture
which is now known as the Hilbert-Waring Theorem.

Any one man can only do so much, so the
greatest mathematicians should help nurture their
colleagues.
Hilbert provided a famous List of 23 Unsolved
Problems, which inspired and directed the development
of 20th-century mathematics.
Hilbert was warmly regarded by his colleagues and students,
and contributed to the careers of several great mathematicians
and physicists including Georg Cantor, Hermann Minkowski,
John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.
His doctoral students included Hermann Weyl, Richard Courant,
Max Dehn, Teiji Takagi, Ernst Zermelo, Wilhelm Ackermann, the chess
champion Emanuel Lasker, and many other famous mathematicians.

Eventually Hilbert turned to physics
and made key contributions to classical and quantum
physics and to general relativity.
He published the Einstein Field Equations
independently of Einstein
(though his writings make clear he treats this as strictly
Einstein's invention).

Minkowski won a prestigious prize at age 18 for
reconstructing Eisenstein's enumeration of the ways to represent
integers as the sum of five squares.
(The Paris Academy overlooked that Smith had already published
a solution for this!)
His proof built on quadratic forms and continued fractions
and eventually led him to the new field of Geometric Number Theory,
for which Minkowski's Convex Body Theorem (a sort of pigeonhole principle) is
often called the Fundamental Theorem.
Minkowski was also a major figure in the development of functional analysis.
With his "question mark function" and "sausage," he was also a
pioneer in the study of fractals.
Several other important results are named after him, e.g.
the Hasse-Minkowski Theorem.
He was first to extend the Separating Axis Theorem to multiple
dimensions.
Minkowski was one of Einstein's teachers, and also a close friend
of David Hilbert.
He is particularly famous for building on
Poincaré's work to invent Minkowski space
to deal with Einstein's Special Theory of Relativity.
This not only provided a better explanation for the Special Theory,
but helped inspire Einstein toward his General Theory.
Minkowski said
that his "views of space and time ... have sprung from the soil
of experimental physics, and therein lies their strength....
Henceforth space by itself, and time by itself, are doomed to fade away
into mere shadows, and only a kind of union of the two will
preserve an independent reality."

Hadamard made revolutionary advances in several different
areas of mathematics, especially complex analysis, analytic number
theory, differential geometry, partial differential equations,
symbolic dynamics, chaos theory, matrix theory, and Markov chains;
for this reason he
is sometimes called the "Last Universal Mathematician."
He also made contributions to physics.
One of the most famous results in mathematics is
the Prime Number Theorem, that there are approximately
n/log n primes less than n.
This result was conjectured by Legendre and Gauss,
attacked cleverly by Riemann and Chebyshev,
and finally, by building on Riemann's work,
proved by Hadamard and Vallee-Poussin.
(Hadamard's proof is considered more elegant and useful than Vallee-Poussin's.)
Several other important theorems are named after
Hadamard (e.g. his Inequality of Determinants),
and some of his theorems are named after others
(Hadamard was first to prove Brouwer's Fixed-Point Theorem
for arbitrarily many dimensions).
Hadamard was also influential in promoting others' work:
He is noted for his survey of Poincaré's work;
his staunch defense of the Axiom of Choice led to the acceptance of Zermelo's
work.
Hadamard was a successful teacher,
with André Weil, Maurice Fréchet, and others acknowledging
him as key inspiration.
Like many great mathematicians he emphasized the importance of intuition, writing
"The object of mathematical rigor is to sanction and
legitimize the conquests of intuition, and there never was any other object for it."

Hausdorff had diverse interests: he composed music
and wrote poetry, studied astronomy, wrote on philosophy,
but eventually focused on mathematics, where he did important work in
several fields including set theory, measure theory,
functional analysis, and both algebraic and point-set topology.
His studies in set theory led him to the
Hausdorff Maximal Principle, and the Generalized Continuum Hypothesis;
his concepts now called Hausdorff measure and Hausdorff dimension led
to geometric measure theory and fractal geometry;
his Hausdorff paradox led directly to the famous Banach-Tarski paradox;
he introduced other seminal concepts, e.g. Hausdorff Distance.
He also worked in analysis, solving the Hausdorff moment problem.

As Jews in Hitler's Germany, Hausdorff and his wife committed
suicide rather than submit to internment.

Cartan worked in the theory of Lie groups
and Lie algebras,
applying methods of topology, geometry and invariant theory to
Lie theory, and classifying all Lie groups.
This work was so significant that
Cartan, rather than Lie, is considered the most important
developer of the theory of Lie groups.
Using Lie theory and ideas like his Method of Prolongation
he advanced the theories of differential equations
and differential geometry.
Cartan introduced several new concepts including
algebraic group, exterior differential forms, spinors,
moving frames, Cartan connections.
He proved several important theorems, e.g. Schläfli's Conjecture
about embedding Riemann metrics, Stokes' Theorem,
and fundamental theorems about symmetric Riemann spaces.
He made a key contribution to Einstein's general relativity,
based on what is now called Riemann-Cartan geometry.
Cartan's methods were so original as to be fully appreciated only recently;
many now consider him to be one of the greatest mathematicians of his era.
In 1938 Weyl called him "the greatest living master in differential geometry."

Borel exhibited great talent while still in his teens,
soon practically founded modern measure theory, and received
several honors and prizes.
Among his famous theorems is the Heine-Borel Covering Theorem.
He also did important work in several other fields of mathematics,
including divergent series, quasi-analytic functions,
differential equations, number theory, complex analysis,
theory of functions, geometry, probability theory, and game theory.
Relating measure theory to probabilities, he
introduced concepts like normal numbers and the Borel-Kolmogorov paradox.
He also did work in relativity and the philosophy of science.
He anticipated the concept of chaos, inspiring Poincaré.
Borel combined great creativity with strong analytic power;
however he was especially interested in applications, philosophy, and
education, so didn't pursue the tedium of rigorous development and proof;
for this reason his great importance as a theorist is often underestimated.
Borel was decorated for valor in World War I, entered politics
between the Wars, and joined the French
Resistance during World War II.

Levi-Civita was noted for strong geometrical intuition,
and excelled at both pure mathematics and mathematical physics.
He worked in analytic number theory, differential
equations, tensor calculus, hydrodynamics, celestial mechanics,
and the theory of stability.
Several inventions are named after him, e.g.
the non-archimedean Levi-Civita field, the Levi-Civita parallelogramoid,
and the Levi-Civita symbol.
His work inspired all three of the greatest 20th-century mathematical
physicists, laying key mathematical groundwork for Weyl's unified field
theory, Einstein's relativity, and Dirac's quantum theory.

Lebesgue did groundbreaking work in real
analysis, advancing Borel's measure theory;
his Lebesgue integral superseded the Riemann integral
and improved the theoretical basis for Fourier analysis.
Several important theorems are named after
him, e.g. the Lebesgue Differentiation Theorem
and Lebesgue's Number Lemma.
He did important work on Hilbert's 19th Problem, and in
the Jordan Curve Theorem for higher dimensions.
In 1916, the Lebesgue integral was compared "with a modern Krupp gun,
so easily does it penetrate barriers which were impregnable."
In addition to his seminal contributions to measure theory
and Fourier analysis, Lebesgue made significant contributions in
several other fields including complex analysis,
topology, set theory, potential theory, dimension theory,
and calculus of variations.

Landau was one of the most prolific
and influential number theorists ever and wrote the first
comprehensive treatment of analytic number theory.
He was also adept at complex function theory.
He was especially keen at finding very simple proofs:
one of his most famous results was a simpler proof
of Hadamard's prime number theorem; being simpler it was also more fruitful
and led to Landau's Prime Ideal Theorem.
In addition to simpler proofs of existing theorems, new theorems by Landau
include important facts about Riemann's Hypothesis;
facts about Dirichlet series;
key lemmas of analysis;
a result in Waring's Problem;
a generalization of the Little Picard Theorem;
and a partial proof of Gauss' conjecture about the density of classes
of composite numbers.
In 1912 Landau described four conjectures about prime numbers which were
'unattackable with present knowledge': (a) Goldbach's conjecture,
(b) infinitely many primes n^2+1, (c) infinitely many twin primes (p, p+2),
(d) a prime exists in every interval (n^2, n^2+n). By 2018 none of these conjectures
have been resolved, though much progress has been made in each case.
Landau was the inventor of big-O notation.
Hardy wrote that no one was ever more passionately devoted to
mathematics than Landau.

Hardy was an extremely prolific research mathematician who
did important work in analysis (especially the theory of integration),
number theory, global analysis, and analytic number theory.
He proved several important theorems about numbers, for example
that Riemann's zeta function has infinitely many zeros
with real part 1/2.
He was also an excellent teacher and wrote several
excellent textbooks, as well as a famous treatise on
the mathematical mind.
He abhorred applied mathematics, treating mathematics as a creative art;
yet his work has found application in
population genetics, cryptography, thermodynamics and
particle physics.

Hardy is especially famous (and important)
for his encouragement of and collaboration
with Ramanujan.
Hardy provided rigorous proofs for several of Ramanujan's
conjectures, including Ramanujan's "Master Theorem" of analysis.
Among other results of this collaboration was the
Hardy-Ramanujan Formula for partition enumeration, which
Hardy later used as a model to develop the Hardy-Littlewood
Circle Method;
Hardy then used this method to prove stronger versions
of the Hilbert-Waring Theorem, and in prime number theory;
the method has continued to be a very productive
tool in analytic number theory.
Hardy was also a mentor to Norbert Wiener, another famous prodigy.

Hardy once wrote "A mathematician, like a painter or poet,
is a maker of patterns.
If his patterns are more permanent than theirs, it is because
they are made with ideas."
He also wrote "Beauty is the first test; there is no permanent place in
the world for ugly mathematics."

Maurice Fréchet introduced the concept of metric spaces
(though not using that term); and also made major contributions
to point-set topology.
Building on work of Hadamard and Volterra,
he generalized Banach spaces to use new (non-normed) metrics
and proved that many
important theorems still applied in these more general spaces.
For this work, and his invention of the notion of compactness,
Fréchet is called the Founder of the Theory of Abstract Spaces.
He also did important work in probability theory and in analysis;
for example he proved the Riesz Representation Theorem the same year Riesz did.
Many theorems and inventions are named after him,
for example Fréchet Distance, which has many applications
in applied math, e.g. protein structure analysis.

Albert Einstein was unquestionably one of the
two greatest physicists in all of history.
The atomic theory achieved general acceptance only after Einstein's
1905 paper which showed that atoms' discreteness explained Brownian motion.
Another 1905 paper introduced the famous equation
E = mc2; yet Einstein published other papers
that same year, two of which were more important and influential
than either of the two just mentioned.
No wonder that physicists speak of the Miracle Year
without bothering to qualify it as Einstein's Miracle Year!
(Before his Miracle Year, Einstein had been a mediocre
undergraduate, and held minor jobs including patent
examiner.)
Altogether Einstein published at least 300 books or papers on physics.
For example, in a 1917 paper he anticipated the principle of the laser.
Also, sometimes in collaboration with Leo Szilard, he was co-inventor of
several devices, including a gyroscopic compass,
hearing aid, automatic camera and, most famously,
the Einstein-Szilard refrigerator.
He became a very famous and influential public figure.
(For example, it was his letter that led Roosevelt to start
the Manhattan Project.)
Among his many famous quotations is:
"The search for truth is more precious than its possession."

Einstein is most famous for his Special and General Theories
of Relativity, but he should be considered the key pioneer of
Quantum Theory as well, drawing inferences from Planck's work that
no one else dared to draw.
Indeed it was his articulation of the quantum principle in a 1905 paper
which has been called
"the most revolutionary sentence written by a physicist of the
twentieth century."
Einstein's discovery of the photon in that paper led to his only Nobel Prize;
years later, he was first to call attention to the "spooky"
nature of quantum entanglement.
Einstein was also first to call attention to a flaw in Weyl's
earliest unified field theory.
But despite the importance of his other contributions it is Einstein's
General Theory which is his most profound contribution.
Minkowski had developed a flat 4-dimensional space-time to
cope with Einstein's Special Theory; but it was Einstein
who had the vision to add curvature to that space
to describe acceleration.

Einstein certainly has the breadth, depth, and historical importance
to qualify for this list; but his genius and significance were not in
the field of pure mathematics.
(He acknowledged his limitation, writing
"I admire the elegance of your [Levi-Civita's] method of computation;
it must be nice to ride through these fields upon the horse of
true mathematics while the like of us have
to make our way laboriously on foot.")
Einstein was a mathematician, however;
he pioneered the application of tensor calculus to physics
and invented the Einstein summation notation.
That Einstein's equation explained a discrepancy in Mercury's
orbit was a discovery made by Einstein personally
(a discovery he described as 'joyous excitement' that gave
him heart palpitations).
He composed a beautiful essay about mathematical proofs
using the Theorem of Menelaus as his example.
Certainly he belongs on a Top 100 List:
his extreme greatness overrides his focus away from math.
Einstein ranks #10 on Michael Hart's famous list of
the Most Influential Persons in History.
His General Theory of Relativity has been called the
most creative and original scientific theory ever.
Einstein once wrote "... the creative principle resides in mathematics
[; thus]
I hold it true that pure thought can grasp reality, as the ancients dreamed."

Oswald Veblen's first mathematical achievement was a
novel system of axioms for geometry.
He also worked in topology; projective geometry; differential geometry
(where he was first to introduce the concept of differentiable manifold);
ordinal theory (where he introduced the Veblen hierarchy); and
mathematical physics where he worked with spinors and relativity.
He developed a new theory of ballistics during World War I
and helped plan the first American computer during World War II.
His famous theorems include the Veblen-Young Theorem
(an important algebraic fact about projective spaces);
a proof of the Jordan Curve Theorem more rigorous than Jordan's;
and Veblen's Theorem itself (a generalization
of Euler's result about cycles in graphs).
Veblen, a nephew of the famous economist Thorstein Veblen,
was an important teacher; his famous students included Alonzo Church,
John W. Alexander, Robert L. Moore, and J.H.C. Whitehead.
He was also a key figure in establishing
Princeton's Institute of Advanced Study;
the first five mathematicians he hired for the Institute were
Einstein, von Neumann, Weyl, J.W. Alexander and Marston Morse.

Brouwer is often considered the "Father of Topology;"
among his important theorems were the Fixed Point Theorem,
the "Hairy Ball" Theorem,
the Jordan-Brouwer Separation Theorem,
and the Invariance of Dimension.
He developed the method of simplicial approximations,
important to algebraic topology;
he also did work in geometry, set theory, measure theory,
complex analysis and the foundations of mathematics.
He was first to anticipate forms like the Lakes of Wada,
leading eventually to other measure-theory "paradoxes."
Several great mathematicians, including Weyl, were inspired
by Brouwer's work in topology.

Brouwer is most famous as the founder of Intuitionism,
a philosophy of mathematics in sharp contrast
to Hilbert's Formalism, but Brouwer's philosophy also
involved ethics and aesthetics and has been compared with
those of Schopenhauer and Nietzsche.
Part of his mathematics thesis was rejected as
"... interwoven with some kind of pessimism and mystical attitude
to life which is not mathematics ..."
As a young man, Brouwer spent a few years to develop
topology, but once his great talent was demonstrated
and he was offered prestigious professorships, he devoted
himself to Intuitionism, and acquired a reputation as
eccentric and self-righteous.

Intuitionism has had a significant influence,
although few strict adherents.
Since only constructive proofs are permitted, strict adherence
would slow mathematical work.
This didn't worry Brouwer who once wrote:
"The construction itself is an art, its application to the world
an evil parasite."

Noether was an innovative researcher
who was considered the greatest master of abstract algebra ever;
her advances included a new theory of ideals, the inverse Galois problem,
and the general theory of commutative rings.
She originated novel reasoning methods, especially one based
on "chain conditions," which advanced invariant theory
and abstract algebra; her insistence on generalization led to a unified
theory of modules and Noetherian rings.
Her approaches tended to unify disparate areas (algebra, geometry,
topology, logic) and led eventually to modern category theory.
Her invention of Betti homology groups led to algebraic topology, and thus
revolutionized topology.

Noether's work has found various applications in physics,
and she made direct advances in mathematical physics herself.
Noether's Theorem establishing that certain symmetries imply
conservation laws has been
called the most important Theorem in physics since the Pythagorean Theorem.
Several other important theorems are named after her, e.g.
Noether's Normalization Lemma, which provided an important
new proof of Hilbert's Nullstellensatz.
Noether was an unusual and inspiring teacher; her successful students
included Emil Artin, Max Deuring, Jacob Levitzki, etc.
She was generous with students and colleagues, even allowing them to claim
her work as their own.
Noether was close friends with the other greatest mathematicians of her
generation: Hilbert, von Neumann, and Weyl.
Weyl once said he was embarrassed to accept the famous Professorship
at Göttingen because Noether was his "superior as a mathematician."
Emmy Noether is considered the greatest female mathematician ever.

Sierpinski won a gold medal as an undergraduate by
making a major improvement to a famous theorem by Gauss
about lattice points inside a circle.
He went on to do important research in set theory, number theory,
point set topology, the theory of functions, and fractals.
He was extremely prolific, producing 50 books and over 700 papers.
He was a Polish patriot: he contributed to the development of Polish
mathematics despite that his land was controlled by Russians or Nazis for most
of his life. He worked as a code-breaker during the Polish-Soviet War,
helping to break Soviet ciphers.

Sierpinski was first to prove Tarski's remarkable conjecture that
the Generalized Continuum Hypothesis implies the Axiom of Choice.
He developed three famous fractals: a space-filling curve;
the Sierpinski gasket; and the Sierpinski carpet, which covers the plane
but has area zero and has found application in antennae design.
Borel had proved that almost all real numbers are "normal" but Sierpinski
was the first to actually display a number which is normal in every base.
He proved the existence of infinitely many Sierpinski numbers having the
property that, e.g. (78557·2n+1) is composite number for
every natural number n. It remains an unsolved problem (likely to
be defeated soon with high-speed computers) whether 78557
is the smallest such "Sierpinski number."

Lefschetz was born in Russia, educated as an engineer
in France, moved to U.S.A., was severely handicapped in an accident,
and then switched to pure mathematics.
He was a key founder of algebraic topology,
even coining the word topology,
and pioneered the application of topology to algebraic geometry.
Starting from Poincaré's work, he developed
Lefschetz duality and used it to derive conclusions about fixed points
in topological mappings.
The Lefschetz Fixed-point Theorem left Brouwer's famous result as just
a special case.
His Picard-Lefschetz theory eventually led to the proof of the Weil conjectures.
Lefschetz also did important work in algebraic geometry,
non-linear differential equations, and control theory.
As a teacher he was noted for a combative style.
Preferring intuition over rigor, he once told a student who had improved on
one of Lefschetz's proofs:
"Don't come to me with your pretty proofs. We don't bother with that
baby stuff around here."

Birkhoff is one of the greatest native-born American mathematicians
ever, and did important work in many fields.
There are several significant theorems named after him:
the Birkhoff-Grothendieck Theorem is an important result about vector
bundles;
Birkhoff's Theorem is an important result in algebra;
and Birkhoff's Ergodic Theorem is a key result in statistical mechanics
which has since been applied to many other fields.
His Poincaré-Birkhoff Fixed Point Theorem is especially important
in celestial mechanics, and led to instant worldwide fame:
the great Poincaré had described it as most important,
but had been unable to complete the proof.
In algebraic graph theory, he invented Birkhoff's
chromatic polynomial (while trying to prove the four-color map theorem);
he proved a significant result in general relativity which implied the
existence of black holes;
he also worked in differential equations and number theory;
he authored an important text on dynamical systems.
Like several of the great mathematicians of that era, Birkhoff
developed his own set of axioms for geometry; it is his axioms
that are often found in today's high school texts.
Birkhoff's intellectual interests went beyond mathematics; he once wrote
"The transcendent importance of love and goodwill in all human relations
is shown by their mighty beneficent effect upon the individual and society."

Weyl studied under Hilbert and became one of the premier
mathematicians and thinkers of the 20th century.
Along with Hilbert and Poincaré he was a great "universal"
mathematician; his discovery of gauge invariance and notion of Riemann
surfaces form the basis of modern physics; he was also
a creative thinker in philosophy.
Weyl excelled at many fields of mathematics including integral equations,
harmonic analysis, analytic number theory, Diophantine approximations,
axiomatic theory, and mathematical philosophy;
but he is most respected for his revolutionary advances
in geometric function theory (e.g., differentiable manifolds),
the theory of compact groups (incl. representation theory), and
theoretical physics (e.g., Weyl tensor, gauge field theory and invariance).
His theorems include key lemmas and foundational results in several fields;
Atiyah commented that whenever
he explored a new topic he found that Weyl had preceded him.
Although he was a master of algebra, he revealed his philosophic
preference by
writing "In these days the angel of topology and the devil of abstract
algebra fight for the soul of every individual discipline of mathematics."
For a while, Weyl was a disciple of Brouwer's Intuitionism and
helped advance that doctrine, but he eventually found it too restrictive.
Weyl was also a very influential figure in all three major fields
of 20th-century physics:
relativity, unified field theory and quantum mechanics.
He and Einstein were great admirers of each other.
Because of his contributions to Schrödinger, many think the latter's
famous result should be named the Schrödinger-Weyl Wave Equation.

Vladimir Vizgin wrote "To this day, Weyl's [unified field]
theory astounds all in the depth of its ideas, its mathematical simplicity,
and the elegance of its realization."
The Nobel prize-winner Julian Schwinger, himself considered an
inscrutable genius, was so impressed by Weyl's book connecting
quantum physics to group theory that he likened Weyl to a "god"
because "the ways of gods are mysterious, inscrutable, and
beyond the comprehension of ordinary mortals."
Weyl once wrote: "My work always tried to unite the Truth
with the Beautiful, but when I
had to choose one or the other, I usually chose the Beautiful."

John Littlewood was a very prolific researcher.
(This fact is obscured somewhat in that many papers
were co-authored with Hardy, and their names were always given
in alphabetic order.)
The tremendous span of his career is suggested by the fact
that he won Smith's Prize (and Senior Wrangler) in 1905
and the Copley Medal in 1958.
He specialized in analysis and analytic number theory
but also did important work in combinatorics, Fourier theory,
Diophantine approximations, differential equations, and other fields.
He also did important work in practical engineering, creating a method
for accurate artillery fire during the First World War,
and developing equations for radio and radar in preparation for the Second War.
He worked with the Prime Number Theorem and Riemann's Hypothesis;
and proved the unexpected fact that Chebyshev's bias, and
Li(x)>π(x), while true for most, and all but
very large, numbers, are violated infinitely often.
(Building on this result, it is now known that there is a big
patch of primes near 109608 that exceed the Li(x)
prediction, though few if any of those primes are actually known.)
Some of his work was elementary, e.g. his elegant proof that a
cube cannot be dissected into unequal cubes; but
most of his results were too specialized to state here, e.g.
his widely-applied
4/3 Inequality which guarantees that certain bimeasures are finite,
and which inspired one of Grothendieck's most famous results.
Hardy once said that his friend was
"the man most likely to storm and smash a really deep and formidable problem;
there was no one else who could command such a combination of insight,
technique and power."
Littlewood's response was that it was possible to be too strong
of a mathematician, "forcing
through, where another might be driven to a different,
and possibly more fruitful, approach."

Like Abel, Ramanujan was a self-taught prodigy who lived in
a country distant from his mathematical peers, and suffered
from poverty: childhood dysentery and vitamin deficiencies probably led
to his early death.
Yet he produced 4000 theorems or conjectures in number theory,
algebra, and combinatorics.
While some of these were old theorems or just curiosities,
many were brilliant new theorems with very difficult proofs.
For example, he found a beautiful identity connecting
Poisson summation to the Möbius function.
He also found a brilliant generalization of Lagrange's
Four Square Theorem; and much more.
Ramanujan might be almost unknown today, except that
his letter caught the eye of Godfrey Hardy, who saw
remarkable, almost inexplicable formulae which
"must be true, because if they were not true, no one would
have had the imagination to invent them."
Ramanujan's specialties included infinite series,
elliptic functions, continued fractions,
partition enumeration, definite integrals, modular equations,
the divisor function,
gamma functions, "mock theta" functions, hypergeometric series,
and "highly composite" numbers.
Ramanujan's "Master Theorem" has wide application in analysis,
and has been applied to the evaluation of Feynman diagrams.
Much of his best work was done in collaboration with Hardy, for example
a proof that almost all numbers n have about log log n
prime factors (a result which developed into probabilistic number theory).
Much of his methodology, including unusual ideas about divergent series,
was his own invention.
(As a young man he made the absurd claim that 1+2+3+4+... = -1/12.
Later it was noticed that this claim translates to a true statement
about the Riemann zeta function, with which Ramanujan was unfamiliar.)
Ramanujan's innate ability for algebraic manipulations equaled or surpassed
that of Euler and Jacobi.

Ramanujan's most famous work was with the partition enumeration
function p(), Hardy guessing that some of these discoveries would have
been delayed at least a century without Ramanujan.
Together, Hardy and Ramanujan developed an analytic approximation to
p(), although Hardy was initially awed by Ramanujan's intuitive
certainty about the existence of such a formula,
and even the form it would have.
(Rademacher and Selberg later discovered an exact
expression to replace the Hardy-Ramanujan approximation;
when Ramanujan's notebooks were studied it
was found he had anticipated their technique, but had
deferred to his friend and mentor.)

In a letter from his deathbed, Ramanujan introduced his mysterious
"mock theta functions",
gave examples, and developed their properties.
Much later these forms began to appear in disparate areas:
combinatorics, the proof of Fermat's Last Theorem, and even
knot theory and the theory of black holes.
It was only recently, more than 80 years after Ramanujan's letter,
that his conjectures
about these functions were proven; solutions mathematicians had sought
unsuccessfully were found among his examples.
Mathematicians are baffled that Ramanujan could make these
conjectures, which they confirmed only with difficulty using methods
not available in Ramanujan's day.

Many of Ramanujan's results
are so inspirational that there is a periodical dedicated to them.
The theories of strings and crystals have benefited from Ramanujan's work.
(Today some professors achieve fame just by finding
a new proof for one of Ramanujan's many results.)
Unlike Abel, who insisted on rigorous proofs, Ramanujan
often omitted proofs.
(Ramanujan may have had unrecorded proofs, poverty leading him to
use chalk and erasable slate rather than paper.)
Unlike Abel, much of whose work depended on the complex
numbers, most of Ramanujan's work focused on real numbers.
Despite these limitations, some consider Ramanujan to be the
greatest mathematical genius ever; but he ranks as low as #17
since many lesser mathematicians were much more influential.

Thoralf Skolem proved fundamental theorems of lattice theory,
proved the Skolem-Noether Theorem of algebra, also worked with set theory
and Diophantine equations; but is best known for his work
in logic, metalogic, and non-standard models.
Some of his work preceded similar results by Gödel.
He developed a theory of recursive functions which anticipated
some computer science.
He worked on the famous Löwenheim-Skolem Theorem which
has the "paradoxical" consequence
that systems with uncountable sets can have countable models.
("Legend has it that Thoralf Skolem, up until the end of his life,
was scandalized by the association of his name to a result of this type,
which he considered an absurdity, nondenumerable sets being, for him,
fictions without real existence.")

George Pólya (Pólya György)
did significant work in several fields: complex analysis,
probability, geometry, algebraic number theory, and combinatorics,
but is most noted for his teaching How to Solve It,
the craft of problem posing and proof.
He is also famous for the Pólya Enumeration Theorem.
Several other important theorems he proved include
the Pólya-Vinogradov Inequality of number theory,
the Pólya-Szego Inequality of functional analysis, and
the Pólya Inequality of measure theory.
He introduced the Hilbert-Pólya Conjecture that the
Riemann Hypothesis might be a consequence of spectral theory
(in 2017 this Conjecture was partially proved by a team of
physicists, and the Riemann Hypothesis might be close
to solution!).
He introduced the famous "All horses are the same color" example
of inductive fallacy; he named the Central Limit Theorem of statistics.
Pólya was the "teacher par excellence": he wrote top
books on multiple subjects; his successful students
included John von Neumann.
His work on plane symmetry groups directly inspired Escher's drawings.
Having huge breadth and influence, Pólya has been called
"the most influential mathematician of the 20th century."

Stefan Banach was a self-taught mathematician
who is most noted as the "Founder of Functional Analysis"
and for his contributions to measure theory.
Among several important theorems bearing his name
are the Uniform Boundedness (Banach-Steinhaus) Theorem,
the Open Mapping (Banach-Schauder) Theorem,
the Contraction Mapping (Banach fixed-point) Theorem,
and the Hahn-Banach Theorem.
Many of these theorems are of practical value to
modern physics; however he also proved
the paradoxical Banach-Tarski Theorem, which demonstrates a
sphere being rearranged into two spheres of the
same original size.
(Banach's proof uses the Axiom of Choice and is sometimes
cited as evidence that that Axiom is false.)
The wide range of Banach's work is indicated by the
Banach-Mazur results in game theory (which also challenge the
axiom of choice).
Banach also made brilliant contributions to probability theory,
set theory, analysis and topology.

Banach once said "Mathematics is the most beautiful and most
powerful creation of the human spirit."

Norbert Wiener entered college at age 11,
studying various sciences;
he wrote a PhD dissertation at age 17 in philosophy of mathematics where
he was one of the first to show a definition of ordered pair as a set.
(Hausdorff also proposed such a definition; both Wiener's
and Hausdorff's definitions have been superceded by Kuratowski's
(a, b) = {{a}, {a, b}} despite that it leads
to a singleton when a=b.)
He then did important work in several topics in applied mathematics, including
stochastic processes (beginning with Brownian motion),
potential theory, Fourier analysis,
the Wiener-Hopf decomposition useful for solving
differential and integral equations,
communication theory, cognitive science, and quantum theory.
Many theorems and concepts are named after him, e.g the Wiener Filter
used to reduce the error in noisy signals.
His most important contribution to pure mathematics was his
generalization of Fourier theory into generalized harmonic analysis,
but he is most famous for his writings on feedback
in control systems, for which he coined the new word, cybernetics.
Wiener was first to relate information to thermodynamic entropy,
and anticipated the theory of information attributed to Claude Shannon.
He also designed an early analog computer.
Although they differed dramatically in both personal and mathematical
outlooks, he and John von Neumann were the two key
pioneers (after Turing) in computer science.
Wiener applied his cybernetics to draw conclusions about human
society which, unfortunately, remain largely unheeded.

Carl Siegel became famous when his doctoral dissertation
established a key result in Diophantine approximations.
He continued with contributions
to several branches of analytic and algebraic number theory,
including arithmetic geometry and quadratic forms.
He also did seminal work with Riemann's zeta function,
Dedekind's zeta functions,
transcendental number theory, discontinuous groups,
the 3-body problem in celestial mechanics,
and symplectic geometry.
In complex analysis he developed Siegel modular forms, which have
wide application in math and physics.
He may share credit with Alexander Gelfond for the
solution to Hilbert's 7th Problem.
Siegel admired the "simplicity and honesty" of masters like Gauss, Lagrange and
Hardy and lamented the modern "trend for senseless abstraction."
He and Israel Gelfand were the
first two winners of the Wolf Prize in Mathematics.
Atle Selberg called him a "devastatingly impressive" mathematician who
did things that "seemed impossible."
André Weil declared that Siegel was the greatest mathematician
of the first half of the 20th century.

Aleksandrov worked in set theory, metric spaces and
several fields of topology, where he developed techniques of
very broad application.
He pioneered the studies of compact and bicompact spaces,
and homology theory. He laid the groundwork for a key theorem
of metrisation.
His most famous theorem may be his discovery about "perfect subsets"
when he was just 19 years old. Much of his work was done in
collaboration with Pavel Uryson and Heinz Hopf.
Aleksandrov was an important teacher; his students included Lev Pontryagin.

Artin was an important and prolific researcher in
several fields of algebra, including algebraic number theory,
the theory of rings, field theory,
algebraic topology, Galois theory,
a new method of L-series, and geometric algebra.
Among his most famous theorems were Artin's Reciprocity Law,
key lemmas in Galois theory,
and results in his Theory of Braids.
He also produced two very influential conjectures:
his conjecture about the zeta function
in finite fields developed into the field of arithmetic geometry;
Artin's Conjecture on primitive roots inspired much work
in number theory, and was later generalized to become Weil's Conjectures.
He is credited with solution to Hilbert's 17th Problem
and partial solution to the 9th Problem.
His prize-winning students include John Tate and Serge Lang.
Artin also did work in physical sciences, and was an accomplished musician.

Dirac had a severe father and was bizarrely taciturn
(perhaps autistic), but became one of the greatest mathematical physicists ever.
He developed Fermi-Dirac statistics, applied quantum theory
to field theory, predicted the existence of magnetic monopoles,
and was first to note that some quantum equations
lead to inexplicable infinities.
His most important contribution was to combine relativity and
quantum mechanics by developing, with pure thought, the Dirac Equation.
From this equation, Dirac deduced the existence of anti-electrons,
a prediction considered so bizarre it was ignored --
until anti-electrons were discovered in a cloud chamber four years later.
For this work he was awarded the Nobel Prize in Physics at age 31,
making him one of the youngest Laureates ever.
Dirac's mathematical formulations, including his Equation and the
Dirac-von Neumann axioms, underpin all of modern particle physics.
After his great discovery, Dirac continued to do important work,
some of which underlies modern string theory.
He was also adept at more practical physics; although he declined an invitation
to work on the Manhattan Project, he did contribute a fundamental
result in centrifuge theory to that Project.

The Dirac Equation was one of the most important scientific
discoveries of the 20th century and Dirac was certainly a
superb mathematical genius, but I've left Dirac off of the Top 100 since
his work didn't advance "pure" mathematics.
Like many of the other greatest mathematical physicists
(Kepler, Einstein, Weyl), Dirac thought the true equations
of physics must have beauty, writing
"... it is more important to have beauty in one's
equations than to have them fit experiment ... [any discrepancy may]
get cleared up with further development of the theory."

Alfred Tarski (born Alfred Tajtelbaum)
was one of the greatest and most prolific logicians ever,
but also made advances in set theory, measure theory, topology,
algebra, group theory, computability theory, metamathematics, and geometry.
He was also acclaimed as a teacher.
Although he achieved fame at an early age with the Banach-Tarski
Paradox, his greatest achievements were in formal logic.
He wrote on the definition of truth, developed model theory, and
investigated the completeness questions which also intrigued Gödel.
He proved several important systems
to be incomplete, but also established completeness results for
real arithmetic and geometry.
His most famous result may be
Tarski's Undefinability Theorem, which is related to
Gödel's Incompleteness Theorem but more powerful.
Several other theorems, theories and paradoxes are named after
Tarski including Tarski-Grothendieck Set Theory,
Tarski's Fixed-Point Theorem of lattice theory (from which
the famous Cantor-Bernstein-Schröder Theorem is a simple corollary),
and a new derivation of the Axiom of Choice
(which Lebesgue refused to publish because "an implication between two
false propositions is of no interest").
Tarski was first to enunciate the remarkable fact that the
Generalized Continuum Hypothesis implies the Axiom of Choice,
although proof had to wait for Sierpinski.
Tarski's other notable accomplishments include his cylindrical
algebra, ordinal algebra,
universal algebra, and an elegant and novel axiomatic basis of geometry.

John von Neumann (born Neumann Janos Lajos)
was a childhood prodigy who could do
very complicated mental arithmetic at an early age.
As an adult he was noted for hedonism and reckless driving
but also became one of the most prolific geniuses in history,
making major contributions in many branches of
both pure and applied mathematics.
He was an essential pioneer of both quantum physics and computer science.

Von Neumann pioneered the use of models in set theory,
thus improving the axiomatic basis of mathematics.
He proved a generalized spectral theorem sometimes called
the most important result in operator theory.
He developed von Neumann Algebras.
He was first to state and prove the Minimax Theorem and
thus invented game theory; this work also advanced operations research;
and led von Neumann to propose the Doctrine of Mutual Assured
Destruction which was a basis for Cold War strategy.
He developed cellular automata (first invented by Stanislaw Ulam),
famously constructing a self-reproducing automaton.
He worked in mathematical foundations: he formulated
the Axiom of Regularity and invented elegant definitions for the
counting numbers (0 = {}, n+1 = n ∪ {n}).
He also worked in analysis, matrix theory,
measure theory, numerical analysis, ergodic theory, group representations,
continuous geometry, statistics and topology.
Von Neumann discovered an ingenious area-conservation paradox
related to the famous Banach-Tarski volume-conservation paradox.
He inspired some of Gödel's famous work
(and independently proved Gödel's Second Theorem).
He is credited with (partial) solution to
Hilbert's 5th Problem using the Haar Theorem;
this also relates to quantum physics.
George Pólya once said
"Johnny was the only student I was ever afraid of.
If in the course of a lecture I stated an unsolved problem,
the chances were he'd come to me as soon as the lecture was over,
with the complete solution in a few scribbles on a slip of paper."
Michael Atiyah has said he calls only three people geniuses:
Wolfgang Mozart, Srinivasa Ramanujan, and Johnny von Neumann.

Von Neumann did very important work in fields other than
pure mathematics.
By treating the universe as a very high-dimensional phase space,
he constructed an elegant mathematical basis
(now called von Neumann algebras)
for the principles of quantum physics.
He advanced philosophical questions about time and
logic in modern physics.
He played key roles in the design of conventional, nuclear and thermonuclear
bombs; he also advanced the theory of hydrodynamics.
He applied game theory and Brouwer's Fixed-Point Theorem
to economics, becoming a major figure in that field.
His contributions to computer science are many:
in addition to co-inventing the stored-program computer,
he was first to use pseudo-random number generation,
finite element analysis, the merge-sort algorithm,
a "biased coin" algorithm, and (though Ulam first conceived the
approach) Monte Carlo simulation.
By implementing wide-number software he joined several other
great mathematicians (Archimedes, Apollonius, Liu Hui, Hipparchus,
Madhava, and (by proxy), Ramanujan)
in producing the best approximation to π of his time.
At the time of his death, von Neumann was working on
a theory of the human brain.

Kolmogorov had a powerful intellect and excelled in
many fields.
As a youth he dazzled his teachers by constructing
toys that appeared to be "Perpetual Motion Machines."
At the age of 19, he achieved fame by finding a Fourier series
that diverges almost everywhere, and decided to devote
himself to mathematics.
He is considered the founder of the fields of intuitionistic logic,
algorithmic complexity theory,
and (by applying measure theory) modern probability theory.
He also excelled in topology, set theory, trigonometric series,
and random processes.
He and his student Vladimir Arnold proved the surprising
Superposition Theorem, which not only solved Hilbert's 13th Problem, but
went far beyond it.
He and Arnold also developed the "magnificent" Kolmogorov-Arnold-Moser
(KAM) Theorem,
which quantifies how strong a perturbation must be to upset
a quasiperiodic dynamical system.
Kolmogorov's axioms of probability are considered a partial solution of
Hilbert's 6th Problem.
He made important contributions to the constructivist ideas
of Kronecker and Brouwer.
While Kolmogorov's work in probability theory had direct
applications to physics, Kolmogorov also did work in physics directly,
especially the study of turbulence.
There are dozens of notions named after Kolmogorov,
such as the Kolmogorov Backward Equation,
the Chapman-Kolmogorov equations,
the Borel-Kolmogorov Paradox,
and the intriguing Zero-One Law of "tail events" among random variables.

Henri Cartan, son of the great Élie Cartan,
is particularly noted for his work in algebraic topology, and
analytic functions; but also worked with sheaves, and many
other areas of mathematics.
He was a key member of the Bourbaki circle.
(That circle was led by Weil,
emphasized rigor, produced important texts, and introduced
terms like in-, sur-, and bi-jection, as well
as the Ø symbol.)
Working with Samuel Eilenberg (also a Bourbakian),
Cartan advanced the theory of homological algebra.
He is most noted for his many contributions
to the theory of functions of several complex variables.
Henri Cartan was an important influence on Grothendieck
and others, and an excellent teacher;
his students included Jean-Pierre Serre.

Gödel, who had the nickname Herr Warum ("Mr. Why")
as a child, was perhaps the foremost logic theorist
ever, clarifying the relationships between various modes
of logic. He partially resolved both Hilbert's 1st and 2nd Problems,
the latter with a proof so remarkable that it was connected to the
drawings of Escher and music of Bach in the title of a famous book.
He was a close friend of Albert Einstein, and was first to discover
"paradoxical" solutions (e.g. time travel) to Einstein's equations.
About his friend, Einstein later said that he had remained at
Princeton's Institute for Advanced Study merely
"to have the privilege of walking home with Gödel."
(Like a few of the other greatest 20th-century
mathematicians, Gödel was very eccentric.)

Two of the major questions confronting mathematics are:
(1) are its axioms consistent (its theorems all being
true statements)?,
and (2) are its axioms complete (its true statements all being theorems)?
Gödel turned his attention to these fundamental questions.
He proved that first-order logic was indeed complete, but that
the more powerful axiom systems needed for arithmetic (constructible
set theory) were necessarily incomplete.
He also proved that the Axioms of Choice (AC) and the Generalized Continuum
Hypothesis (GCH) were consistent with set theory, but that set
theory's own consistency could not be proven.
He may have established that the truths of AC and GCH were independent
of the usual set theory axioms, but the proof was left to Paul Cohen.

In Gödel's famous proof of Incompleteness,
he exhibits a true statement (G) which
cannot be proven, to wit
"G (this statement itself) cannot be proven."
If G could be proven it would be a contradictory
true statement, so consistency dictates that it indeed cannot
be proven. But that's what G says, so G is true!
This sounds like mere word play, but building from ordinary logic
and arithmetic Gödel was able to construct statement G rigorously.

Weil made profound contributions to several areas of mathematics,
especially algebraic geometry, which he showed to have deep connections
with number theory.
His Weil conjectures were very influential; these and other
works laid the groundwork for some of Grothendieck's work.
Weil proved a special case of the Riemann Hypothesis; he contributed,
at least indirectly, to the recent proof of Fermat's Last Theorem;
he also worked in group theory, general and algebraic topology,
differential geometry, sheaf theory, representation theory, and
theta functions.
He invented several new concepts including vector bundles,
and uniform space.
His work has found applications in particle physics and string theory.
He is considered to be one of the most influential of
modern mathematicians.

Weil's biography is interesting. He studied Sanskrit as a child,
loved to travel, taught at a Muslim university in India for
two years (intending to teach French civilization),
wrote as a young man under the famous pseudonym
Nicolas Bourbaki, spent time in prison during World War II
as a Jewish objector, was almost executed as a spy, escaped to
America, and eventually joined Princeton's
Institute for Advanced Studies.
He once wrote:
"Every mathematician worthy of the name has experienced [a]
lucid exaltation in which one thought succeeds another as if miraculously."

Shiing-Shen Chern (Chen Xingshen) studied under
Élie Cartan, and became perhaps the greatest
master of differential geometry.
He is especially noted for his work in algebraic geometry, topology
and fiber bundles, developing his Chern characters
(in a paper with "a tremendous number of geometrical jewels"),
developing Chern-Weil theory,
the Chern-Simons invariants,
and especially for his brilliant generalization of
the Gauss-Bonnet Theorem to multiple dimensions.
His work had a major influence in several fields of
modern mathematics as well as gauge theories of physics.
Chern was an important influence in China and a highly
renowned and successful teacher:
one of his students (Yau) won the Fields Medal,
another (Yang) the Nobel Prize in physics.
Chern himself was the first Asian to win the prestigious Wolf Prize.

Turing developed a new foundation for mathematics
based on computation;
he invented the abstract Turing machine,
designed a "universal" version of such a machine,
proved the famous Halting Theorem (related to
Gödel's Incompleteness Theorem), and
developed the concept of machine intelligence
(including his famous Turing Test proposal).
He also introduced the notions of definable number and
oracle (important in modern computer science),
and was an early pioneer in the study of neural networks.
For this work he is called the Father
of Computer Science and Artificial Intelligence.
Turing also worked in group theory, numerical analysis,
and complex analysis; he developed an
important theorem about Riemann's zeta function;
he had novel insights in quantum physics.
During World War II he turned his talents to cryptology;
his creative algorithms were considered possibly
"indispensable" to the decryption of German Naval Enigma coding,
which in turn is judged to have certainly shortened the War by at
least two years.
Although his clever code-breaking algorithms were
his most spectacular contributions at Bletchley Park,
he was also a key designer of the Bletchley "Bombe" computer.
After the war he helped design other physical computers,
as well as theoretical designs;
and helped inspire von Neumann's later work.
He (and earlier, von Neumann) wrote about the Quantum Zeno
Effect which is sometimes called the Turing Paradox.
He also studied the mathematics of biology,
especially the Turing Patterns of morphogenesis
which anticipated the discovery of BZ reactions.
Turing's life ended tragically:
charged with immorality and forced to undergo chemical
castration, he apparently took his own life.
With his outstanding depth and breadth, Alan Turing would
qualify for our list in any event, but his decisive contribution to the war
against Hitler gives him unusually strong historic importance.

Erdös was a childhood prodigy who became a
famous (and famously eccentric) mathematician.
He is best known for work in combinatorics (especially Ramsey Theory)
and partition calculus, but made contributions
across a very broad range of mathematics, including
graph theory, analytic number theory, probabilistic methods,
and approximation theory.
He is regarded as the second most prolific mathematician in history,
behind only Euler.
Although he is widely regarded as an important and influential
mathematician, Erdös founded no new field of mathematics:
He was a "problem solver" rather than a "theory developer."
He's left us several still-unproven intriguing conjectures, e.g.
that 4/n = 1/x + 1/y + 1/z has positive-integer solutions
for any n.

Erdös liked to speak of "God's Book of Proofs" and discovered new,
more elegant, proofs of several existing theorems,
including the two most famous and important about prime numbers:
Chebyshev's Theorem that there is always a prime
between any n and 2n, and
(though the major contributor was Atle Selberg)
Hadamard's Prime Number Theorem itself.
He also proved many new theorems, such as
the Erdös-Szekeres Theorem about monotone subsequences
with its elegant (if trivial) pigeonhole-principle proof.

Eilenberg is considered a founder of category theory,
but also worked in algebraic topology, automata theory and other areas.
He coined several new terms including functor,
category, and natural isomorphism.
Several other concepts are named after him, e.g. a proof
method called the Eilenberg telescope
or Eilenberg-Mazur Swindle.
He worked on cohomology theory, homological algebra, etc.
By using his category theory and axioms of homology, he unified
and revolutionized topology.
Most of his work was done in collaboration with others, e.g.
Henri Cartan; but he also single-authored an important text laying
a mathematical foundation for theories of computation and language.
Sammy Eilenberg was also a noted art collector.

Gelfand was a brilliant and important mathematician
of outstanding breadth with a huge number of theorems and discoveries.
He was a key figure of functional analysis and integral geometry;
he pioneered representation theory, important to modern physics;
he also worked in many fields of analysis,
soliton theory, distribution theory, index theory, Banach algebra,
cohomology, etc.
He made advances in physics and biology as well as mathematics.
He won the Order of Lenin three times and several prizes from
Western countries.
Considered one of the two greatest Russian mathematicians of
the 20th century,
the two were compared with
"[arriving in a mountainous country]
Kolmogorov would immediately try to climb the highest mountain;
Gelfand would immediately start to build roads."
In old age Israel Gelfand emigrated to the U.S.A. as a professor,
and won a MacArthur Fellowship.

Shannon's initial fame was for a
paper called "possibly the most important master's thesis of the century."
That paper founded digital circuit design theory by proving that universal
computation was achieved with an ensemble of switches and boolean gates.
He also worked with analog computers, theoretical genetics, and sampling
and communication theories.
Early in his career Shannon was fortunate to work with several other
great geniuses including Weyl, Turing, Gödel and even Einstein;
this may have stimulated him toward a broad range of interests and expertise.
He was an important and prolific inventor, discovering signal-flow graphs,
the topological gain formula, etc.; but also inventing the first
wearable computer (to time roulette wheels in Las Vegas casinos),
a chess-playing algorithm, a flame-throwing trumpet,
and whimsical robots (e.g. a "mouse" that navigated a maze).
His hobbies included juggling, unicycling, blackjack card-counting.
His investigations into gambling theory led to new approaches to the stock market.

Shannon worked in cryptography during World War II; he was first to
note that a one-time pad allowed unbreakable encryption as long as the pad was
as large as the message; he is also noted for Shannon's maxim that a
code designer should assume the enemy knows the system.
His insights into cryptology eventually led to
information theory, or the mathematical theory of communication, in which
Shannon established the relationships among bits, entropy, power and noise.
It is as the Founder of Information Theory that Shannon has become immortal.

Selberg may be the greatest analytic number theorist ever.
He also did important work in
Fourier spectral theory,
lattice theory (e.g. introducing and partially
proving the conjecture that "all lattices are arithmetic"),
and the theory of automorphic forms, where he introduced
Selberg's Trace Formula.
He developed a very important result in analysis called
the Selberg Integral.
Other Selberg techniques of general utility include
mollification, sieve theory, and the Rankin-Selberg method.
These have inspired other mathematicians, e.g.
contributing to Deligne's proof of the Weil conjectures.
Selberg is also famous for
ground-breaking work on Riemann's Hypothesis, and
the first "elementary" proof of the Prime Number Theorem.

Tate, a student of Emil Artin, is a master of algebraic
number theory, p-adic theory and arithmetic geometry.
Using Fourier analysis and Tate cohomology groups, he
revolutionized the treatments of class field theory and
algebraic K-theory.
In addition to Tate cohomology groups, Tate's key inventions
include rigid analytic geometry, Hodge-Tate theory, Tate-Barsotti groups,
applications of adele ring self-duality, the
Tate module, Tate curve, Tate twists, and much more.
His long and productive career earned the Abel Prize
for his "vast and lasting impact on the theory of numbers
[and] his incisive contributions and illuminating insights ...
He has truly left a conspicuous imprint on modern mathematics."

Serre did important work with spectral sequences
and algebraic methods,
revolutionizing the study of algebraic topology and algebraic geometry,
especially homotopy groups and sheaves.
Hermann Weyl praised Serre's work strongly, saying it
gave an important new algebraic basis to analysis.
He collaborated with Grothendieck and Pierre Deligne, helped resolve the
Weil conjectures, and contributed indirectly to
the recent proof of Fermat's Last Theorem.
His wide range of research areas also includes
number theory, bundles, fibrations, p-adic modular forms,
Galois representation theory, and more.
Serre has been much honored: he is the youngest ever to win
a Fields Medal; 49 years after his Fields Medal he became
the first recipient of the Abel Prize.

Grothendieck has done brilliant work in several areas of mathematics
including number theory, geometry, topology,
and functional analysis,
but especially in the fields of algebraic geometry
and category theory, both of which he revolutionized.
He is especially noted for his invention of the Theory of Schemes,
and other methods to unify different branches of mathematics.
He applied algebraic geometry to number theory;
applied methods of topology to set theory; etc.
Grothendieck is considered a master of abstraction, rigor and presentation.
He has produced many important and deep results in homological algebra,
most notably his etale cohomology.
With these new methods, Grothendieck and his outstanding student Pierre Deligne
were able to prove the Weil Conjectures.
Grothendieck also developed the theory of sheafs, the theory of motives,
generalized the Riemann-Roch Theorem to revolutionize K-theory,
developed Grothendieck categories, crystalline cohomology,
infinity-stacks and more.
The guiding principle behind much of Grothendieck's work has been
Topos Theory, which he invented to harness the methods of topology.
These methods and results have redirected several diverse branches
of modern mathematics including number theory, algebraic topology,
and representation theory.
Among Grothendieck's famous results was his Fundamental Theorem
in the Metric Theory of Tensor Products, which was inspired by
Littlewood's proof of the 4/3 Inequality.

Grothendieck's radical religious and political philosophies
led him to retire from public life while still in his prime, but
he is widely regarded as the greatest mathematician of the 20th
century, and indeed one of the greatest geniuses ever.

The Riemann Embedding Problems were important puzzles
of geometry that baffled many of the greatest minds for a century.
Hilbert showed that Lobachevsky's hyperbolic plane
could not be embedded into Euclidean 3-space, but what about into
Euclidean 4-space?
Cartan and Chern were among the great mathematicians who solved
various special cases, but using "methods entirely without precedent"
John Nash demonstrated a general solution.
This was a true highlight of 20th-century mathematics.

Nash was a lonely, tormented schizophrenic whose life
was portrayed in the film Beautiful Mind.
He achieved early fame with his work in game theory; this eventually
led to the Nobel Prize in Economics.
Earlier studies in game theory focused on the simplest
cases (two-person zero-sum, or cooperative), but Nash
demonstrated "Nash equilibria" for n-person or non-zero-sum
non-cooperative games.
Nash also excelled at several other fields of mathematics,
especially topology, algebraic geometry, partial differential equations,
elliptic functions, and the theory of manifolds (including
singularity theory, the concept of real algebraic manifolds
and isotropic embeddings).
He proved theorems of great importance which had defeated all
earlier attempts.
His most famous theorems were the Nash Embedding Theorems,
e.g. that any Riemannian manifold of dimension k can be embedded
isometrically into some n-dimensional Euclidean space.
Other important work was in partial differential equations where
he proved that strong regularity constraints apply to
solutions of the equations of heat and fluid flow.

Carleson is a master of complex analysis, especially
harmonic analysis, and dynamical systems; he proved many difficult
and important theorems;
among these are a theorem about
quasiconformal mapping extension, a technique to construct higher
dimensional strange attractors,
and the famous Kakutani Corona Conjecture, whose proof brought
Carleson great fame.
For the Corona proof he introduced Carleson measures, one of several useful
tools he's created for his masterful proofs.
In 1966, four years after proving Kakutani's Conjecture,
he proved the 53-year old Luzin's Conjecture,
a strong statement about Fourier convergence.
This was startling because of a 38-year old conjecture suggested
by Kolmogorov that Luzin's Conjecture was false.

Atiyah's career has had extraordinary breadth and depth.
He advanced the theory of vector bundles;
this developed into topological K-theory
and the Atiyah-Singer Index Theorem.
This Index Theorem is considered one of the most
far-reaching theorems ever, subsuming famous old results (Descartes'
total angular defect, Euler's topological characteristic),
important 19th-century theorems (Gauss-Bonnet, Riemann-Roch),
and incorporating important work by Weil and especially Shiing-Shen Chern.
It is a key to the study of high-dimension
spaces, differential geometry, and equation solving.
Several other key results are named after Atiyah,
e.g. the Atiyah-Bott Fixed-Point Theorem,
the Atiyah-Segal Completion Theorem,
and the Atiyah-Hirzebruch spectral sequence.
Atiyah's work developed important connections not only between
topology and analysis, but with modern physics; Atiyah himself has
been a key figure in the development of string theory;
and is a proponent of the recent idea that octonions may
underlie particle physics.
He has also studied the physics of instantons and monopoles.
This work, and Atiyah-inspired work in gauge theory, restored
a close relationship between leading edge research
in mathematics and physics.
His interest in physics, and an old theory of von Neumann,
have led him, as a very old man, to explore the fine structure
constant of physics.
In September 2018 he announced that this
recent work would lead to a proof of the Riemann Hypothesis
but other mathematicians are quite skeptical.
Nonetheless, Michael Atiyah is still regarded as one of the
very greatest mathematicians of the 20th century.

Atiyah is known as a vivacious genius in person, inspiring many,
e.g. Edward Witten.
Atiyah once said a mathematician must sometimes "freely float in the
atmosphere like a poet and imagine the whole universe of possibilities,
and hope that eventually you come down to Earth somewhere else."
He also said "Beauty is an important criterion in mathematics
... It determines what you regard as important and what is not."

Milnor founded the field of differential topology
and has made other major advances in topology,
algebraic geometry and dynamical systems.
He discovered Milnor maps (related to fiber bundles);
important theorems in knot theory;
the Duality Theorem for Reidemeister Torsion;
the Milnor Attractors of dynamical systems;
a new elegant proof of Brouwer's "Hairy Ball" Theorem; and much more.
He is especially famous for two counterexamples which each
revolutionized topology.
His "exotic" 7-dimensional hyperspheres
gave the first examples of homeomorphic manifolds that were not also
diffeomorphic, and developed the fields of
differential topology and surgery theory.
Milnor invented certain high-dimensional polyhedra to disprove the
Hauptvermutung ("main conjecture") of geometric topology.
While most famous for his exotic counterexamples,
his revolutionary insights into dynamical systems have important value to
practical applied mathematics.
Although Milnor has been called the "Wizard of Higher Dimensions,"
his work in dynamics began with novel insights into very
low-dimensional systems.

As Fields, Presidential and (twice) Putnam Medalist, as well as
winner of the Abel, Wolf and two Steele Prizes; Milnor
can be considered the most "decorated" mathematician of the modern era.

Roger Penrose is a thinker of great breadth, who has contributed
to biology and philosophy, as well as to mathematics, general relativity and
cosmology.
Some of his earliest work was done in collaboration with his father Lionel,
a polymath and professor of psychiatry who developed the Penrose Square Root Law of
voting theory.
Together, Roger and his father discovered the 'impossible tri-bar' and
an impossible staircase which inspired work by the artist M.C. Escher.
And, in turn, Escher's drawings may have helped inspire Penrose's
most famous discoveries in recreational mathematics: non-periodic tilings.
He soon found such a tiling with just two tile shapes;
the previous record was six shapes.
(Nine years after that, such tilings were observed in nature as "quasi-crystals.")
Penrose has written several successful popular books on science.

As a mathematician, Penrose did important work in algebra: he was first to conceive
of the generalized matrix inverse, and used it for novel solutions in linear
algebra and spectral decomposition.
He did more important work in geometry and topology; for example, he proved
theorems about embedding (or "unknotting") manifolds in Euclidean space.
His best mathematics, e.g. the invention of twistor theory, was inspired
by his pursuit of Einstein's general relativity.

Penrose is most noted for his very creative work in cosmology, specifically
in the mathematics of gravitation, space-time, black holes and the Big Bang.
He developed new methods to apply spinors and Riemann tensors to gravitation.
His twistor theory was an effort to relate general relativity
to quantum theory; this work advanced both physics and mathematics.
The top physicist Kip Thorne said "Roger Penrose revolutionised the
mathematical tools that we use to analyse the properties of space-time."
Stephen Hawking was an early convert to Penrose's methods; the mathematical
laws of black holes (and the Big Bang)
are called the Penrose-Hawking Singularity Theorems.
Penrose formulated the Censorship Hypotheses about black holes, e.g. the Riemannian
Penrose Inequality and the Weyl Curvature Hypothesis;
and discovered Penrose-Terrell rotation.

Penrose has proposed conformal cyclic cosmology, that in
the entropy death of one universe, the scaling of time and distance
become arbitrary and the dying universe becomes the big bang for another.
Recently it is proposed
that evidence for this can be seen in the details of
the cosmic microwave background radiation from the early universe.
Many of his theories are extremely controversial:
He claims that Gödel's
Incompleteness Theorem provides insight into human consciousness.
He has developed a detailed theory that quantum effects (involving
the microtubules in neurons) enhance the capability of biologic brains.

Thompson is the master of finite groups. He achieved
early fame by proving a long-standing conjecture about Frobenius groups.
He followed up by proving (with Walter Feit) that all nonabelian
finite simple groups are of even order. This result, proved in
a 250-page paper, stunned the world of mathematics; it led to
the classification of all finite groups.
Thompson also made major contributions to coding theory,
and to the inverse Galois problem.
His work with Galois groups has been called "the most important
advance since Hilbert's time."

Langlands started by studying semigroups and partial differential
equations but soon switched his attention to representation theory where he
found deep connections between group theory and
automorphic forms; he then used these connections to make profound
discoveries in number theory.
Langlands' methods, collectively called the Langlands Program,
are now central to all of these fields.
The Langlands Dual Group LG revolutionized representation theory
and led to a large number of conjectures.
One of these conjectures is the Principle of Functoriality, of which a partial
proof allowed Langlands to prove a famous conjecture of Artin, and Wiles
to prove Fermat's Last Theorem.
Langlands and others have applied these methods to prove several
other old conjectures, and to formulate new more powerful conjectures.
He has also worked with Eisenstein series, L-functions, Lie groups,
percolation theory, etc.
He mentored several important mathematicians (including Thomas Hales,
mentioned briefly in Pappus' mini-bio).

Langlands once wrote "Certainly the best times were when I was alone with
mathematics, free of ambition and pretense, and indifferent to the world."
He was appointed Hermann Weyl Professor at the Institute for Advanced Study
and now sits in the office once occupied by Albert Einstein.
This seems appropriate since, as the man "who reinvented mathematics,"
his advances have sometimes been compared to Einstein's.

Arnold is most famous for solving Hilbert's 13th Problem;
for the "magnificent" Kolmogorov-Arnold-Moser (KAM) Theorem;
and for "Arnold diffusion," which identifies exceptions to the
stability promised by the KAM Theorem.
In addition to dynamical systems theory, Arnold made contributions
to catastrophe theory, topology, algebraic geometry, symplectic geometry,
differential equations, mathematical physics; and was
the essential founder of modern singularity theory.

Conway has done pioneering work in a very broad range
of mathematics including knot theory, number theory, group theory,
lattice theory, combinatorial game theory, geometry, quaternions,
tilings, and cellular automaton theory.
He started his career by proving a case of Waring's Problem,
but achieved fame when he discovered the largest then-known
sporadic group (the symmetry group of the Leech lattice);
this sporadic group is now known to be second in size
only to the Monster Group, with which Conway also worked.
Conway's fertile creativity has produced a
cornucopia of fascinating inventions: markable straight-edge
construction of the regular heptagon (a feat also achieved
by Alhazen, Thabit, Vieta and perhaps Archimedes),
a nowhere-continuous function that has the Intermediate Value property,
the Conway box function,
the rational tangle theorem in knot theory,
the aperiodic pinwheel tiling,
a representation of symmetric polyhedra,
the silly but elegant Fractran programming language,
his chained-arrow notation for large numbers, and many results and
conjectures in recreational mathematics.
He found the simplest proof for Morley's Trisector Theorem (sometimes
called the best result in simple plane geometry since ancient Greece).
He proved an unusual theorem about quantum physics:
"If experimenters have free will, then so do elementary particles."
His most famous construction is
the computationally complete automaton
known as the Game of Life.
His most important theoretical invention, however,
may be his surreal numbers incorporating infinitesimals;
he invented them to solve combinatorial games like Go,
but they have pure mathematical significance as the
largest possible ordered field.

Conway's great creativity and breadth
certainly make him one of the greatest living mathematicians.
Conway has won the Nemmers Prize in Mathematics,
and was first winner of the Pólya Prize.

Gromov is considered one of the greatest
geometers ever, but he has a unique "soft" approach to geometry
which leads to applications in other fields: Gromov has contributed to
group theory, partial differential equations,
other areas of analysis and algebra, and even mathematical biology.
He is especially famous for his pseudoholomorphic curves;
they revolutionized the study of symplectic manifolds and are important
in string theory.
By applying his geometric ideas to all areas of mathematics,
Gromov has become one of the most influential living mathematicians.
He has proved a very wide variety of theorems: important results
about groups of polynomial growth, theorems essential to Perelman's proof
of the Poincaré conjecture, the nonsqueezing theorem
of Hamiltonian mechanics, theorems of systolic geometry,
and various inequalities and compactness theorems.
Several concepts are named after him, including Gromov-Hausdorff convergence,
Gromov-Witten invariants, Gromov's random groups, Gromov product, etc.

Using new ideas about cohomology, in 1974 Pierre Deligne
stunned the world of mathematics with a spectacular proof of the Weil
conjectures. Proof of these conjectures, which were key to further progress
in algebraic geometry, had eluded the great Alexandre Grothendieck.
With his "unparalleled blend of penetrating insights,
fearless technical mastery and dazzling ingenuity," Deligne made other
important contributions to a broad range of mathematics in addition to
algebraic geometry, including algebraic and analytic number theory,
topology, group theory, the Langlands conjectures,
Grothendieck's theory of motives, and Hodge theory.
Deligne also found a partial solution of Hilbert's 21st Problem.
Several ideas are named after him including
Deligne-Lusztig theory, Deligne-Mumford stacks, Fourier-Deligne transform,
the Langlands-Deligne local constant, Deligne cohomology, and at least
eight distinct conjectures.

Shelah has advanced logic, model theory, set theory,
and especially the theory of cardinal numbers.
His work has led to new methods in diverse fields like group theory,
topology, measure theory, stability theory,
algebraic geometry,
Banach spaces, and combinatorics.
He solved outstanding problems like Morley's Problem; proved
the independence of the Whitehead problem; found
a "Jonsson group" and counterexamples to outstanding conjectures;
improved on Arrow's Voting Theorem; and, most famously,
proved key results about singular cardinals.

Shelah is the founder of the theories of proper forcing, classification
of models, and possible cofinalities.
He has authored over 800 papers and several books,
making him one of the most prolific mathematicians in history.
He has been described as a "phenomenal mathematician, ... produc[ing]
results at a furious pace."

Thurston revolutionized the study of 3-D manifolds;
it was his work which eventually led Perelman
to a proof of Poincaré's Conjecture.
He developed the key results of foliation theory and, with Gromov,
co-founded geometric group theory.
One of his award citations states "Thurston has fantastic geometric
insight and vision: his ideas have completely revolutionised the study
of topology in 2 and 3 dimensions, and brought about a new and fruitful
interplay between analysis, topology and geometry."

Witten is the world's greatest living physicist, and one of the top
mathematicians. Not only does Witten apply mathematics to solve problems in
physics, but his broad knowledge of physics, especially quantum field
theory, string theory, supersymmetry, and quantum gravity,
has led him to novel connections and insights in abstract geometry and topology,
as well as physics.
His skill with string theory led him to a novel theory of invariants and
allowed him to improve results in knot theory;
his skill with supersymmetry led him to new results in differential geometry.
He has applied quantum field theory to higher-dimensional spaces and
found new insights there.
He has proven several important new theorems of mathematics and
general relativity but also has had unproven
insights which inspired proofs by others.
His discovery that the five competing models of string theory were all
congruent to a single 'M-theory' (sometimes called a 'Theory of Everything')
revolutionized string theory.
Several theorems or concepts are named after Witten, including
Seiberg-Witten theory, the Weinberg-Witten theorem, the Gromov-Witten invariant,
the Witten index, Witten conjecture, Witten-type Topological quantum field theory, etc.

Witten started his college career studying fields like history and linguistics.
When he finally switched to math and physics he learned at breathtaking speed.
His fellows do not compare him to other living mathematical physicists; they
compare him to Einstein, Weyl, Newton and Ramanujan.

Tao was a phenomenal child prodigy who has become perhaps
the greatest living mathematician. He has made important contributions
to partial differential equations, combinatorics, harmonic analysis,
number theory, group theory, model theory, nonstandard analysis,
random matrices,
the geometry of 3-manifolds, functional analysis, ergodic theory, etc.
and areas of applied math including quantum mechanics,
general relativity, and image processing.
He has been called the first since David Hilbert to
be expert across the entire spectrum of mathematics.
Among his earliest important discoveries were results about the
multi-dimensional Kakeya needle problem, which led to advances in
Fourier analysis and fractals.
In addition to his numerous research papers he has written many highly
regarded textbooks.
One of his prize citations commends his "sheer technical power,
his other-worldly ingenuity for hitting upon new ideas,
and a startlingly natural point of view."

Paul Erdös mentored Tao when he was a ten-year old prodigy,
and the two are frequently compared. They are both prolific
problem solvers across many fields, though have founded no new fields.
As with Erdös, much of Tao's work has been done in collaboration:
for example with Van Vu he proved the circular law of random matrices;
with Ben Green he proved the Dirac-Motzkin conjecture and
solved the "orchard-planting problem."
Especially famous is the Green-Tao Theorem that there are arbitrarily long
arithmetic series among the prime numbers (or indeed among any sufficiently
dense subset of the primes). This confirmed an old
conjecture by Lagrange, and was especially remarkable because the
proof fused methods from number theory, ergodic theory, harmonic analysis,
discrete geometry, and combinatorics.
Tao is also involved in recent efforts to attack the
famous Twin Prime Conjecture.